Formulae [3]
| y = f(x) | dy/dx = f′(x) |
|---|---|
| c (Constant) | 0 |
| xⁿ | n xⁿ⁻¹ |
| \[\frac{1}{x}\] | \[-\frac{1}{x^2}\] |
| \[\frac{1}{x^n}\] | \[-\frac{n}{x^{n+1}}\] |
| \[\sqrt{x}\] | \[\frac{1}{2\sqrt{x}}\] |
| sin x | cos x |
| cos x | −sin x |
| tan x | sec² x |
| sec x | sec x tan x |
| cosec x | −cosec x cot x |
| cot x | −cosec² x |
| eˣ | eˣ |
| aˣ | aˣ log a |
| log x | \[\frac{1}{x}\] |
| logₐ x | \[\frac{1}{x\log a}\] |
| 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 |
Theorems and Laws [1]
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)`
Key Points
Let f(x), g(x) and h(x) are three real valued functions are given, then
(i) Sum of two functions
\[[f(x)+g(x)]^{\prime}=f^{\prime}(x)+g^{\prime}(x)\]
(ii) Difference of two functions
\[[f(x)-g(x)]^{\prime}=f^{\prime}(x)-g^{\prime}(x)\]
(iii) Product of two or more functions
(a) \[[f(x)\cdot g(x)]^{\prime}=f^{\prime}(x)\cdot g(x)+f(x)\cdot g^{\prime}(x)\]
(b) \[\frac{d}{dx}[f(x)\cdot g(x)\cdot h(x)]=f(x)\cdot g(x)\cdot h^{\prime}(x)\]\[+f\left(x\right)\cdot g^{\prime}\left(x\right)\cdot h(x)+f^{\prime}\left(x\right)\cdot g\left(x\right)\cdot h\left(x\right)\]
(iv) Quotient of two functions
\[\left[\frac{f(x)}{g(x)}\right]'\] = \[\frac{f^{\prime}(x)\cdot g(x)-g^{\prime}(x)\cdot f(x)}{\left[g(x)\right]^{2}}\]
provided g(x) ≠ 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., \[[f(x)]^{g(x)}\]
- If an equation contains both x and y and cannot be solved directly for y, it is called an implicit function.
- Implicit functions are generally written in the form:
f(x, y) = 0 - To differentiate an implicit function, differentiate both sides with respect to x, treating y as a function of x.
If x = f(t) and y = g(t) are differentiable functions of parameter t, then
\[\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{\left(\frac{\mathrm{d}y}{\mathrm{d}t}\right)}{\left(\frac{\mathrm{d}x}{\mathrm{d}t}\right)},\frac{\mathrm{d}x}{\mathrm{d}t}\neq0\]
- If y = f(x), then \[\frac{dy}{dx}\] = f′(x) is called the first-order derivative.
- The derivative of the first derivative is called the second-order derivative:
\[\frac{d^2y}{dx^2}\] = f″(x) - Higher order derivatives are written as:
fⁿ(x) or \[\frac{d^ny}{dx^n}\]
Concepts [10]
- Introduction & Derivatives of Some Standard Functions
- Algebra of Differentiation
- Derivative of Composite Functions
- Geometrical Meaning of Derivative
- Derivative of Inverse Function
- Logarithmic Differentiation
- Derivative of Implicit Functions
- Derivative of Parametric Functions
- Higher Order Derivatives
- Successive Differentiation
