Definitions [9]
A function f(x) is said to be continuous at a point x = a, if the following three conditions are satisfied
- f is defined at every point on an open interval containing a.
- \[\lim_{x\to a}f\left(x\right)\] exists.
- \[\lim_{x\to a}f\left(x\right)=f\left(a\right)\].
The principle which states that for a non-viscous liquid in streamline flow passing through a tube of varying cross-section, the product of the area of cross-section and the velocity of flow remains constant at every point is called the Equation of Continuity.
A function f(x) is said to be discontinuous at x = a if it is not continuous at x = a, i.e.
- \[\lim_{x\to a}f\left(a\right)\] does not exist.
- The left-hand limit and the right-hand limit are not equal.
- \[\lim_{x\to a}f\left(x\right)\neq f\left(a\right)\].
If \[\lim_{x\to a^{-}}f\left(x\right)=\lim_{x\to a^{+}}f\left(x\right)\neq f\left(a\right),\] then f(x) is said to be removable discontinuous.
If \[\lim_{x\to a^{+}}f\left(x\right)\neq\lim_{x\to a^{-}}f\left(x\right),\] then f(x) is said to be non-removable discontinuous.
The set of all points where a function is continuous is called its domain of continuity.
A function f(x) is said to be differentiable at x = a if Rf'(a) and Lf'(a) both exist and are equal; it is said to be non-differentiable.
Let f(x) be a real function and a be a point in its domain.
A function f is continuous at x = a iff all three conditions hold:
- f(a) is defined
- \[\lim_{x\to a}f(x)\] exists
- \[\lim_{x\to a}f(x)\] = f(a)
\[\lim_{x\to a^-}f(x)=\lim_{x\to a^+}f(x)=f(a)\]
For the open interval:
A function f is said to be continuous on an open interval (a, b) if it is continuous at every point in the interval.
For a closed interval:
A function f is said to be continuous on the closed interval [a,b] iff:
-
f is continuous at every point of (a,b)
-
f is right continuous at a
\[\lim_{x\to a^+}f(x)=f(a)\] -
f is left continuous at b
\[\lim_{x\to b^-}f(x)=f(b)\]
Formulae [14]
For a non-viscous liquid in streamline flow passing through a tube of varying cross-section:
av = constant
or equivalently:
a ∝ \[\frac {1}{v}\]
where:
- a = area of cross-section of the tube
- v = velocity of flow of the liquid
| Function | Derivative |
|---|---|
| [f(x)]ⁿ | n[f(x)]ⁿ⁻¹ · f′(x) |
| \[\sqrt{\mathrm{f}(x)}\] | \[\frac{1}{2\sqrt{\mathrm{f}(x)}}\cdot\mathrm{f}^{\prime}(x)\] |
| \[\frac{1}{\mathrm{f}(x)}\] | \[-\frac{1}{\left[\mathrm{f}(x)\right]^{2}}\cdot\mathrm{f}^{\prime}(x)\] |
| sin(f(x)) | cos(f(x)) · f′(x) |
| cos(f(x)) | −sin(f(x)) · f′(x) |
| tan(f(x)) | sec²(f(x)) · f′(x) |
| cot(f(x)) | −cosec²(f(x)) · f′(x) |
| sec(f(x)) | sec(f(x)) tan(f(x)) · f′(x) |
| cosec(f(x)) | −cosec(f(x)) cot(f(x)) · f′(x) |
| \[\mathbf{a}^{\mathbf{f}(x)}\] | \[a^{f(x)}\log a\cdot f^{\prime}(x)\] |
| \[\mathrm{e}^{\mathrm{f}(x)}\] | \[\mathrm{e}^{\mathrm{f}(x)\cdot\mathrm{f}^{\prime}(x)}\] |
| log(f(x)) | \[\frac{1}{\mathrm{f}(x)}\cdot\mathrm{f}^{\prime}(x)\] |
| logₐ(f(x)) | \[\frac{1}{\mathrm{f}(x)\mathrm{loga}}\cdot\mathrm{f}^{\prime}(x)\] |
| Function | Derivative | Condition |
|---|---|---|
| sin⁻¹x | \[\frac{1}{\sqrt{1-x^{2}}}\] | |x| < 1 |
| sin⁻¹(f(x)) | \[\frac{1}{\sqrt{1-\{f\left(x\right)\}^{2}}}\frac{d}{dx}f\left(x\right)\] | |f(x)| < 1 |
| cos⁻¹x | \[-\frac{1}{\sqrt{1-x^{2}}}\] | x| < 1 |
| cos⁻¹(f(x)) | \[-\frac{1}{\sqrt{1-\left\{f\left(x\right)\right\}^{2}}}\frac{d}{dx}f(x)\] | |f(x)| < 1 |
| tan⁻¹x | \[\left(\frac{1}{1+x^{2}}\right)\] | x ∈ R |
| tan⁻¹(f(x)) | \[\frac{1}{1+\left\{f\left(x\right)\right\}^{2}}\frac{d}{dx}f(x)\] | f(x) ∈ R |
| cot⁻¹x | \[-\left(\frac{1}{1+x^{2}}\right)\] | x ∈ R |
| cot⁻¹(f(x)) | \[-\frac{1}{1+\{f(x)\}^{2}}\frac{d}{dx}f(x)\] | f(x) ∈ R |
| sec⁻¹x | \[\frac{1}{|x|\sqrt{x^{2}-1}}\] | |x| > 1 |
| sec⁻¹(f(x)) | \[\frac{1}{|f(x)|\sqrt{\{f(x)\}^{2}-1}}\frac{d}{dx}f(x)\] | |f(x)| > 1 |
| cosec⁻¹x | \[-\left(\frac{1}{|x|\sqrt{x^{2}-1}}\right)\] |
|x| > 1 |
| cosec⁻¹(f(x)) | \[-\frac{1}{|f(x)|\sqrt{\{f(x)\}^{2}-1}}\frac{d}{dx}f(x)\] | |f(x)| > 1 |
Left Derivative at x = c:
\[\lim_{h\to0^-}\frac{f(c+h)-f(c)}{h}\]
Right Derivative at x = c
\[\lim_{h\to0^+}\frac{f(c+h)-f(c)}{h}\]
1. Chain Rule:
If u = g(x) and y = f(u), then
\[\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}\]
2. Composite Function Form:
If h(x) = f(g(x)), then
\[h^{\prime}(x)=f^{\prime}(g(x))\cdot g^{\prime}(x)\]
3. Power of a Function:
If y = [f(x)]n, then
\[\frac{dy}{dx}=n[f(x)]^{n-1}\cdot f^{\prime}(x)\]
4. Inverse Function Formula:
\[\frac{dy}{dx}=\frac{1}{\frac{dx}{dy}},\quad\frac{dx}{dy}\neq0\]
\[\frac{dy}{dx}\cdot\frac{dx}{dy}=1\]
5. Derivative of Absolute Value Function:
For y=∣x∣,
\[\frac{d}{dx}(|x|)=\frac{x}{|x|},\quad x\neq0\]
6. Special Results:
\[\frac{d}{dx}(x)=1\]
\[\frac{d}{dx}\left(\frac{1}{x}\right)=-\frac{1}{x^2},x\neq0\]
\[\frac{d}{dx}(\sqrt{x})=\frac{1}{2\sqrt{x}},\mathrm{~}x>0\]
\[\frac{d}{dx}(\sqrt{ax+b})=\frac{a}{2\sqrt{ax+b}}\]
A function f is said to have a derivative at any point x if
\[f^{\prime}(x)=\lim_{h\to0}\frac{f(x+h)-f(x)}{h}\]
\[\frac{d}{dx}(y^n)=ny^{n-1}\frac{dy}{dx}\]
\[\frac{d}{dx}(xy)=x\frac{dy}{dx}+y\]
| Function / Rule | Derivative |
|---|---|
| log x | \[\frac{1}{x}\] |
| \[\log_{a}x\] | \[\frac{1}{x\log a}\] |
| \[\log_ax^n\] | \[n\log_ax\] |
| log u | \[\frac{1}{u}\cdot\frac{du}{dx}\] |
| \[log_a1\] | 0 |
| \[\log_aa\] | 1 |
| \[log_au\] | \[\frac{1}{u\log a}\cdot\frac{du}{dx}\] |
| \[\log_a(xy)\] | \[\log_ax+\log_ay\] |
| \[\log_a\left(\frac{x}{y}\right)\] | \[\log_ax-\log_ay\] |
| \[\log_ax\] | \[\frac{\log x}{\log a}\] |
| \[y=u^{v}\] | \[u^v\frac{d}{dx}(v\log u)\] |
For a 2×2 determinant:
\[F^{\prime}(x)=
\begin{vmatrix}
f_1^{\prime}(x) & f_2(x) \\
g_1^{\prime}(x) & g_2(x)
\end{vmatrix}+
\begin{vmatrix}
f_1(x) & f_2^{\prime}(x) \\
g_1(x) & g_2^{\prime}(x)
\end{vmatrix}\]
For a 3×3 determinant:
\[\mathrm{F^{\prime}}(x)=
\begin{vmatrix}
f_1^{\prime}(x) & f_2^{\prime}(x) & f_3^{\prime}(x) \\
g_1(x) & g_2(x) & g_3^{\prime}(x) \\
h_1(x) & h_2(x) & h_3(x)
\end{vmatrix}+
\begin{vmatrix}
f_1(x) & f_2(x) & f_3(x) \\
g_1^{\prime}(x) & g_2^{\prime}(x) & g_3^{\prime}(x) \\
h_1(x) & h_2(x) & h_3(x)
\end{vmatrix}+
\begin{vmatrix}
f_1(x) & f_2(x) & f_3(x) \\
g_1(x) & g_2(x) & g_3(x) \\
h_1^{\prime}(x) & h_2^{\prime}(x) & h_3^{\prime}(x)
\end{vmatrix}\]
Quotient Rule:
If \[y=\frac{u}{v}\] then \[\frac{dy}{dx}=\frac{v\frac{du}{dx}-u\frac{dv}{dx}}{v^2}\]
Reciprocal Rule:
\[\frac{d}{dx}{\left(\frac{1}{f(x)}\right)}=-\frac{f^{\prime}(x)}{[f(x)]^2}\]
First derivative:
If x = f(t), y = ϕ(t) then \[\frac{dy}{dx}=\frac{\frac{dy}{dt}}{\frac{dx}{dt}}\]
Second derivative:
\[\frac{d^2y}{dx^2}=\frac{\frac{d}{dt}\left(\frac{dy}{dx}\right)}{\frac{dx}{dt}}\]
-
-
-
\[(a^x)^y=a^{xy}\]
- \[a^{-x}=\frac{1}{a^x}\]
\[\frac{d}{dx}(e^x)=e^x\]
\[\frac{d}{dx}(a^x)=a^x\log a,\quad a>0,a\neq1\]
\[\frac{d}{dx}(e^{f(x)})=e^{f(x)}\cdot f^{\prime}(x)\]
\[\frac{d}{dx}(a^{f(x)})=a^{f(x)}\log a\cdot f^{\prime}(x)\]
A. Trigonometric Functions
| Function (y) | \[\frac{dy}{dx}\] |
|---|---|
| sin x | cos x |
| cos x | -sin x |
| tan x | sec2 x |
| cot x | -cosec2 x |
| sec x | sec x.tan x |
| cosec x | -cosec x cot x |
B. Inverse Trigonometric Functions
| Function | Derivative |
|---|---|
| sin−1x | \[\frac{1}{\sqrt{1-x^2}}\] |
| cos−1x | \[-\frac{1}{\sqrt{1-x^2}}\] |
| tan−1x | \[\frac{1}{1+x^2}\] |
| cot−1x | \[-\frac{1}{1+x^2}\] |
| sec−1x | \[\frac{1}{x\sqrt{x^2-1}}\] |
| cosec-1x | \[-\frac{1}{x\sqrt{x^2-1}}\] |
(i) Product of two functions
If y = uv then, \[\frac{dy}{dx}=u\frac{dv}{dx}+v\frac{du}{dx}\]
(i) Product of three functions
If y = uvw then \[\frac{dy}{dx}=uv\frac{dw}{dx}+uw\frac{dv}{dx}+vw\frac{du}{dx}\]
Theorems and Laws [9]
Prove that the function f given by f(x) = |x − 1|, x ∈ R is not differentiable at x = 1.
Any function will not be differentiable if the left-hand limit and the right-hand limit are not equal.
f(x) = |x − 1|, x ∈ R
f(x) = (x − 1), if x − 1 > 0
= −(x − 1), if x − 1 < 0
At x = 1
f(1) = 1 − 1 = 0
left-side limit:
`lim_(h -> 0^-) (f(1 - h) - f(1))/ -h`
= `lim_(h -> 0^-) (1 - (1 - h) - 0)/ (- h)`
= `lim_(h -> 0^-) (+ h)/(- h)`
= −1
Right-side limit:
= `lim_(h -> 0^+) (f(1 + h) - f(1))/h`
= `lim_(h -> 0^+) ((1 + h) - 1 - 0)/ h`
= `lim_(h -> 0^+) h/h`
= 1
Left-side limit and the right-side limit are not equal.
Hence, f(x) is not differentiable at x = 1.
Prove that the greatest integer function defined by f(x) = [x], 0 < x < 3 is not differentiable at x = 1 and x = 2.
Any function will not be differentiable if the left-hand limit and the right-hand limit are not equal.
f(x) = [x], 0 < x < 3
(i) At x = 1
Left-side limit:
`lim_(h -> 0) ([1 - h] - [1])/-h`
= `lim_(h -> 0) (0 - 1)/-h`
= `lim_(h -> 0) 1/h`
= Infinite (∞)
Right-hand limit:
`lim_(h -> 0) ([1 + h] - [1])/h`
= `lim_(h -> 0) (1 - 1)/h`
= 0
Left-side limit and right-side limit are not equal.
Hence, f(x) is not differentiable at x = 1.
(ii) At x = 2
Left-side limit:
`lim_(h -> 0) (f(2 + h) - f(2))/h`
= `lim_(h -> 0) ([2 + h]-2)/h`
= `lim_(h -> 0) (2 -2)/h`
= 0
Right-hand limit:
`lim_(h -> 0) (f(2 - h) - f (2))/h`
= `lim_(h -> 0) ([2 - h] - [2])/-h`
= `lim_(h -> 0) (1 - 2)/-h`
= Infinite (∞)
Left-side limit and right-side limit are not equal.
Hence, f(x) is not differentiable at x = 2.
If y = `[(f(x), g(x), h(x)),(l, m,n),(a,b,c)]`, prove that `dy/dx = |(f'(x), g'(x), h'(x)),(l,m, n),(a,b,c)|`.
y = `|(f(x), g(x), h(x)),(l, m, n),(a, b, c)|`
`dy/dx= |(d/dx (f(x)), d/dx (g(x)), d/dx (h(x))), (l, m, n), (a, b, c)| + |(f(x), g(x), h(x)),(0, 0, 0),(a, b, c)| + |(f(x), g(x), h(x)),(l, m, n),(0, 0, 0)|`
`= |(f'(x), g'(x), h'(x)),(l, m, n),(a, b, c)|`
If (x – a)2 + (y – b)2 = c2, for some c > 0, prove that `[1+ (dy/dx)^2]^(3/2)/((d^2y)/dx^2)` is a constant independent of a and b.
Given, (x – a)2 + (y – b)2 = c2 ...(1)
On differentiating with respect to x,
`=> 2 (x - a) + 2(y - b) dy/dx = 0`
`=> (x - a) + (y - b) dy/dx = 0` ...(2)
Differentiating again with respect to x,
`1 + dy/dx * dy/dx + (y - b) (d^2 y)/dx^2` = 0
`1 + (dy/dx)^2 + (y - b) (d^2y)/dx^2` = 0
`=> (y - b) = - {(1 + (dy/dx)^2)/((d^2y)/dx^2)}` ...(3)
Putting the value of (y – b) in (2),
`(x - a) = {(1 + (dy/dx)^2)/((d^2y)/dx^2)}(dy/dx)` ...(4)
Putting the values of (x − a) and (y − b) from (3) and (4) in (1),
`{1 + (dy/dx)^2}^2/((d^2y)/dx^2)^2 * (dy/dx)^2 + {(1 + (dy/dx)^2)/((d^2y)/dx^2)} = c^2`
On multiplying by `((d^2y)/dx^2)^2`,
`[1 + (dy/dx)^2]^2 (dy/dx)^2 + [1 + (dy/dx)^2]^2 = c^2 ((d^2y)/dx)^2`
`=> [1 + (dy/dx)^2]^2 [(dy/dx)^2 + 1] = c^2 ((d^2y)/dx^2)^2`
`=> {1 + (dy/dx)^2}^3 = c^2 ((d^2y)/dx^2)^2`
On taking the square root,
`therefore {1 + (dy/dx)^2}^(3//2)/((d^2y)/dx^2)` = c ...(a constant independent of a and b.)
If x = `e^(x/y)`, then prove that `dy/dx = (x - y)/(xlogx)`.
Given: x = `e^(x/y)`
Taking log on both the sides,
log x = `log e^(x/y)`
⇒ log x = `x/y log e`
⇒ log x = `x/y` ...[∵ log e = 1] ...(i)
Differentiating both sides w.r.t. x:
`d/dx log x = d/dx (x/y)`
⇒ `1/x = (y xx 1 - x xx dy/dx)/y^2`
⇒ `y^2 = xy - x^2 xx dy/dx`
⇒ `x^2 xx dy/dx = xy - y^2`
⇒ `dy/dx = (y(x - y))/x^2`
⇒ `dy/d = y/x xx ((x - y)/x)`
⇒ `dy/dx = 1/logx xx ((x - y)/x) ...[∵ log x = x/y "from equation (i)"]`
`dy/dx = (x - y)/(xlogx)`
Hence proved.
If cos y = x cos (a + y), with cos a ≠ ± 1, prove that `dy/dx = cos^2(a+y)/(sin a)`.
cos y = x cos (a + y)
∴ x = `(cos y)/(cos (a + y))`
On differentiating with respect to y,
`cos (a + y) d/dy cos y - cos y d/dy`
`therefore dx/dy = (cos (a + y))/(cos^2 (a + y))`
`= (- sin y cos (a + y) + cos y sin (a + y))/(cos^2 (a + y))`
`= (sin (a + y) cos y - cos (a + y) sin y)/(cos^2 (a + y))`
`= (sin (a + y - y))/(cos^2 (a + y))` ...[∵ sin (A − B) = sin A cos B − cos A sin B]
`= (sin a)/(cos^2 (a + y))`
`therefore dy/dx = (cos^2 (a + y))/(sin a)`
If y = 5 cos x – 3 sin x, prove that `(d^2y)/(dx^2) + y = 0`.
Given, y = 5 cos x – 3 sin x
Differentiating both sides with respect to x,
`dy/dx = 5 d/dx cos x - 3 d/dx sin x`
= 5 (−sin x) − 3 cos x
= −5 sin x − 3 cos x
Differentiating both sides again with respect to x,
`(d^2 y)/dx = - 5 d/dx sin x - 3 d/dx cos x`
= −5 cos x − 3 (−sin x)
= 3 sin x − 5 cos x
Hence, `(d^2 y)/dx^2 + y` = 0
(3 sin x − 5 cos x) + (5 cos x − 3 sin x) = 0 ...(On substituting the value of y)
f a function f(x) is
-
Continuous on [a,b]
-
Differentiable on (a,b)
Then there exists at least one c ∈ (a,b) such that
\[f^{\prime}(c)=\frac{f(b)-f(a)}{b-a}\]
If a function f(x) is
-
Continuous on [a,b]
-
Differentiable on (a,b)
-
f(a) = f(b)
Then there exists at least one c ∈ (a,b) such that f′(c) = 0
Key Points
If f & g are two continuous functions at x = a, then
αf is continuous at x = a ∀ α ∈ R
f + g is continuous at x = a
f − g is continuous at x = a
f·g is continuous at x = a
f/g is continuous at x = a, provided g(a) ≠ 0
If y is a differentiable function of u and u is a differentiable function of x, then
\[\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{\mathrm{d}y}{\mathrm{d}u}\cdot\frac{\mathrm{d}u}{\mathrm{d}x}\]
If y = f(x) is a differentiable function of x such that the inverse function x = f⁻¹(y) exists, then x is a differentiable function of y and
\[\frac{\mathrm{d}x}{\mathrm{d}y}=\frac{1}{\left(\frac{\mathrm{d}y}{\mathrm{d}x}\right)}\], where \[\frac{\mathrm{d}y}{\mathrm{d}x}\neq0\].
If differentiation of an expression is done after taking the logarithm on both sides, then it is called logarithmic differentiation. Generally, we apply this method when the given expression is in one of the following forms:
- product of a number of functions,
- a quotient of functions,
- a function which is the power of another function, i.e., \[[f(x)]^{g(x)}\]
A function fails to be continuous at x = a if any one of the following occurs:
-
f(a) is not defined
-
\[\lim_{x\to a}f(x)\] does not exist
-
Either LHL or RHL does not exist
-
Or LHL ≠ RHL
-
-
\[\lim_{x\to a}f(x)\] exists but \[\lim_{x\to a}f(x)\] ≠ f(a)
(a) Left Hand Continuity at x = a
A function is left continuous at x = a if:
-
f(a) exists
-
\[\lim_{x\to a^-}f(x)\mathrm{~exists}\]
-
\[\lim_{x\to a^-}f(x)=f(a)\]
(b) Right Hand Continuity at x = a
A function is right continuous at x = a if:
-
f(a) exists
-
\[\lim_{x\to a^+}f(x)\mathrm{~exists}\]
-
\[\lim_{x\to a^+}f(x)=f(a)\]
c) Continuity at x = a
A function is continuous at x = a iff it is both left continuous and right continuous at x = a.
| Basis of Comparison | Removable Discontinuity | Non-Removable Discontinuity |
|---|---|---|
| Existence of \[\lim_{x\to a}f(x)\] | Exists | Does not exist |
| Left Hand Limit (LHL) | Exists | May not exist |
| Right Hand Limit (RHL) | Exists | May not exist |
| Relation between LHL & RHL | LHL = RHL | LHL ≠ RHL (or one/both do not exist) |
| Value of f(a) | Not defined OR f(a) ≠ \[\lim_{x\to a}f(x)\] | May or may not be defined |
| Continuity at ( x = a ) | Discontinuous | Discontinuous |
| Graphical interpretation | Hole/gap in the graph | Jump, break or vertical asymptote |
| Nature of discontinuity | Temporary | Permanent |
Important Questions [47]
- The function f(x) = x |x| is ______.
- Find the Values of a and B, If the Function F Defined by F ( X ) = { X 2 + 3 X + a , X ⩽ 1 B X + 2 , X > 1 is Differentiable at X = 1.
- The function f(x) = x | x |, x ∈ R is differentiable ______.
- If f(x) = ,,{x2,ifx≥1x,ifx<1, then show that f is not differentiable at x = 1.
- If f(x) = {ax+b;0<x≤12x2-x;1<x<2 is a differentiable function in (0, 2), then find the values of a and b.
- If f(x) = | cos x |, then πf(3π4) is ______.
- If F(X) = X + 1, Find D D X ( F O F ) ( X )
- If Y = ( Sec − 1 X ) 2 , X > 0 Show that X 2 ( X 2 − 1 ) D 2 Y D X 2 + ( 2 X 3 − X ) D Y D X − 2 = 0
- If Sin Y = Xsin(A + Y) Prove that D Y D X = Sin 2 ( a + Y ) Sin a
- If `(X^2 + Y^2)^2 = Xy` Find `(Dy)/(Dx)`
- If tan(x+yx-y) = k, then dydx is equal to ______.
- If `Xpowery + Ypowerx = Apowerb`Then Find `Dybydx`
- If X = E Cos 2 T and Y = E Sin 2 T , Prove that D Y D X = − Y Log X X Log Y .
- If Xy - Yx = Ab, Find D Y D X .
- If x = e^(x/y), then prove that dy/dx = x − y/x log x.
- If Ey ( X +1) = 1, Then Show that D 2 Y D X 2 = ( D Y D X ) 2 .
- If y=log[x+x2+a2] show that (x2+a2)d2ydx2+xdydx=0
- if xx+xy+yx=ab, then find dy/dx.
- Differentiate the function with respect to x. (sin𝑥)^𝑥 + sin^−1√𝑥
- If `Xpowerm Ypowern = (X + Y)Power(M + N)`, Prove that `(Dsquare2y)/(Dxsquare2)= 0`
- If Y = Sin − 1 X + Cos − 1 X , Find D Y D X
- Differentiate the following function with respect to x: (log x)x+x(logx)
- Find D Y D X , If Y = Sin − 1 2 X + 1 1 + 4 X
- If ( Sin X ) Y = X + Y , Find D Y D X
- If y = (log x)x + xlog x, find "dy"/"dx".
- The derivative of x2x w.r.t. x is ______.
- Find the value of dy/dx at θ=pi/4 if x=ae^θ (sinθ-cosθ) and y=ae^θ(sinθ+cosθ)
- If x = a sin 2t (1 + cos 2t) and y = b cos 2t (1 – cos 2t) then find dy/dx
- If x = a sin 2t (1 + cos2t) and y = b cos 2t (1 – cos 2t), find the values of dydxat t = π4
- If x = cos t (3 – 2 cos2 t) and y = sin t (3 – 2 sin2 t), find the value of dx/dy at t =4/π.
- If X = A (2θ – Sin 2θ) And Y = A (1 – Cos 2θ), Find `Dy/Dx` When `Theta = Pi/3`
- If x=a sin 2t(1+cos 2t) and y=b cos 2t(1−cos 2t), find dy/dx
- If x=α sin 2t (1 + cos 2t) and y=β cos 2t (1−cos 2t), show that dy/dx=β/αtan t
- If e^y (x + 1) = 1, show that (d^2y)/(dx^2) = (dy/dx)^2.
- If x = a cos θ + b sin θ, y = a sin θ − b cos θ, show that y2 (d2y)/(dx2)-xdy/dx+y=0
- If x = a cos t and y = b sin t, then find d2ydx2.
- If x cos(a+y)= cosy then prove that dy/dx=(cos^2(a+y)/sina) Hence show that sina(d^2y)/(dx^2)+sin2(a+y)(dy)/dx=0
- If y = ax+b, prove that y(d2ydx2)+(dydx)2 = 0.
- If x = a sin t and y = a (cost+log tan(t/2)) ,find ((d^2y)/(dx^2))
- If x = A cos 4t + B sin 4t, then d2xdt2 is equal to ______.
- If y=2 cos(logx)+3 sin(logx), prove that x^2(d^2y)/(dx2)+x dy/dx+y=0
- If y = tan x + sec x then prove that d2ydx2=cosx(1-sinx)2.
- Differentiate x^sinx+(sinx)^cosx with respect to x.
- Prove that : cos^-1 (12/13) + sin^-1(3/5) = sin^-1(56/65)
- Prove that : 2 Sin − 1 ( 3 5 ) − Tan − 1 ( 17 31 ) = π 4 .
- If x = sin t, y = sin pt, prove that(1-"x"^2)("d"^2"y")/"dx"^2 - "x" "dy"/"dx" + "p"^2"y" = 0
- If Y = Sin (Sin X), Prove that `(D^2y)/(Dx^2) + Tan X Dy/Dx + Y Cos^2 X = 0`
Concepts [16]
- Continuous and Discontinuous Functions
- Algebra of Continuous Functions
- Concept of Differentiability
- Derivative of Composite Functions
- Derivatives of Implicit Functions
- Derivative of Inverse Function
- Exponential and Logarithmic Functions
- Logarithmic Differentiation
- Derivatives of Functions in Parametric Forms
- Second Order Derivative
- Derivative - Exponential and Log
- Proof Derivative X^n Sin Cos Tan
- Infinite Series
- Higher Order Derivative
- Mean Value Theorem
- Overview of Continuity and Differentiability
