Advertisements
Advertisements
प्रश्न
Differentiate log `[(sqrt(1 + x^2) + x)/(sqrt(1 + x^2 - x)]]` w.r.t. cos (log x).
Advertisements
उत्तर
Let y = log `[(sqrt(1 + x^2) + x)/(sqrt(1 + x^2 - x)]]` and v = cos (log x)
Then we want to find `"du"/"dv"`.
u = `log((sqrt(1 + x^2) + x)/(sqrt(1 + x^2 - x)) xx (sqrt(1 + x^2) + x)/(sqrt(1 + x^2 + x)))`
= `log[((sqrt(1 + x^2) + x)^2)/(1 + x^2 - x^2)]`
= `2 log (sqrt(1 + x^2) + x)`
∴ `"du"/"dx" = 2"d"/"dx"[log(sqrt(1 + x^2) + x)]`
= `(2)/(sqrt(1 + x^2) + x)."d"/"dx"(sqrt(1 + x^2) + x)`
= `(2)/(sqrt(1 + x^2) + x).[1/(2sqrt(1 + x^2))."d"/"dx"(1 + x^2) + 1]`
= `(2)/(sqrt(1 + x^2) + x).[(2x)/(2sqrt(1 + x^2)) + 1]`
= `(2)/(sqrt(1 + x^2) + x)(x/sqrt(1 + x^2) + 1)`
= `(2(x + sqrt(1 + x^2)))/((sqrt(1 + x^2) + x)sqrt(1 + x^2)`
= `(2)/sqrt(1 + x^2)`
`"dv"/"dx" = "d"/"dx"[cos(logx)]`
= `-sin(logx)"d"/"dx"(logx)`
= `[-sin(logx)] xx (1)/x`
= `(-sin(logx))/x`
∴ `"du"/"dv" = (("du"/"dx"))/(("dv"/"dx")`
= `(((2)/(sqrt(1 + x^2))))/[[((-sin(logx)))/"x"]`
= `(-2x)/(sqrt(1 + x^2).sin(logx))`.
APPEARS IN
संबंधित प्रश्न
If y=eax ,show that `xdy/dx=ylogy`
Find dy/dx if x sin y + y sin x = 0.
Find `bb(dy/dx)` in the following:
2x + 3y = sin x
Find `bb(dy/dx)` in the following:
x3 + x2y + xy2 + y3 = 81
Show that the derivative of the function f given by
If for the function
\[\Phi \left( x \right) = \lambda x^2 + 7x - 4, \Phi'\left( 5 \right) = 97, \text { find } \lambda .\]
If \[f\left( x \right) = x^3 + 7 x^2 + 8x - 9\]
, find f'(4).
Find the derivative of the function f defined by f (x) = mx + c at x = 0.
Examine the differentialibilty of the function f defined by
\[f\left( x \right) = \begin{cases}2x + 3 & \text { if }- 3 \leq x \leq - 2 \\ \begin{array}xx + 1 \\ x + 2\end{array} & \begin{array} i\text { if } - 2 \leq x < 0 \\\text { if } 0 \leq x \leq 1\end{array}\end{cases}\]
If f (x) = |x − 2| write whether f' (2) exists or not.
If \[\lim_{x \to c} \frac{f\left( x \right) - f\left( c \right)}{x - c}\] exists finitely, write the value of \[\lim_{x \to c} f\left( x \right)\]
Let \[f\left( x \right)\begin{cases}a x^2 + 1, & x > 1 \\ x + 1/2, & x \leq 1\end{cases}\] . Then, f (x) is derivable at x = 1, if
Find `"dy"/"dx"` ; if x = sin3θ , y = cos3θ
Find `"dy"/"dx"` ; if y = cos-1 `("2x" sqrt (1 - "x"^2))`
If x = tan-1t and y = t3 , find `(dy)/(dx)`.
If y = `sqrt(cosx + sqrt(cosx + sqrt(cosx + ... ∞)`, then show that `"dy"/"dx" = sinx/(1 - 2y)`.
Find `"dy"/"dx"`, if : x = sinθ, y = tanθ
Find `"dy"/"dx"` if : x = a cos3θ, y = a sin3θ at θ = `pi/(3)`
Find `"dy"/"dx"` if : x = t2 + t + 1, y = `sin((pit)/2) + cos((pit)/2) "at" t = 1`
DIfferentiate x sin x w.r.t. tan x.
Differentiate `tan^-1((sqrt(1 + x^2) - 1)/(x)) w.r.t tan^-1((2xsqrt(1 - x^2))/(1 - 2x^2))`.
If y = sin (m cos–1x), then show that `(1 - x^2)(d^2y)/(dx^2) - x"dy"/"dx" + m^2y` = 0.
If x = a sin t – b cos t, y = a cos t + b sin t, show that `(d^2y)/(dx^2) = -(x^2 + y^2)/(y^3)`.
Find the nth derivative of the following : y = eax . cos (bx + c)
Choose the correct option from the given alternatives :
If f(x) = `sin^-1((4^(x + 1/2))/(1 + 2^(4x)))`, which of the following is not the derivative of f(x)?
Choose the correct option from the given alternatives :
If y = `tan^-1(x/(1 + sqrt(1 - x^2))) + sin[2tan^-1(sqrt((1 - x)/(1 + x)))] "then" "dy"/"dx"` = ...........
Solve the following :
f(x) = –x, for – 2 ≤ x < 0
= 2x, for 0 ≤ x < 2
= `(18 - x)/(4)`, for 2 < x ≤ 7
g(x) = 6 – 3x, for 0 ≤ x < 2
= `(2x - 4)/(3)`, for 2 < x ≤ 7
Let u (x) = f[g(x)], v(x) = g[f(x)] and w(x) = g[g(x)]. Find each derivative at x = 1, if it exists i.e. find u'(1), v' (1) and w'(1). If it doesn't exist, then explain why?
Differentiate the following w.r.t. x : `tan^-1[sqrt((sqrt(1 + x^2) + x)/(sqrt(1 + x^2) - x))]`
If sin y = x sin (a + y), then show that `"dy"/"dx" = (sin^2(a + y))/(sina)`.
If x = `e^(x/y)`, then show that `dy/dx = (x - y)/(xlogx)`
Find `"dy"/"dx"` if, x3 + x2y + xy2 + y3 = 81
Find `"dy"/"dx"` if, yex + xey = 1
If `x^7 * y^9 = (x + y)^16`, then show that `dy/dx = y/x`
If `"x"^"a"*"y"^"b" = ("x + y")^("a + b")`, then show that `"dy"/"dx" = "y"/"x"`
Find `"dy"/"dx"` if x = `"e"^"3t", "y" = "e"^(sqrt"t")`.
If x2 + y2 = 1, then `(d^2x)/(dy^2)` = ______.
If x = sin θ, y = tan θ, then find `("d"y)/("d"x)`.
State whether the following statement is True or False:
If `sqrt(x) + sqrt(y) = sqrt("a")`, then `("d"y)/("d"x) = 1/(2sqrt(x)) + 1/(2sqrt(y)) = 1/(2sqrt("a"))`
If y = `e^(m tan^-1x)` then show that `(1 + x^2) (d^2y)/(dx^2) + (2x - m) (dy)/(dx)` = 0
If y = y(x) is an implicit function of x such that loge(x + y) = 4xy, then `(d^2y)/(dx^2)` at x = 0 is equal to ______.
If 2x + 2y = 2x+y, then `(dy)/(dx)` is equal to ______.
Let y = y(x) be a function of x satisfying `ysqrt(1 - x^2) = k - xsqrt(1 - y^2)` where k is a constant and `y(1/2) = -1/4`. Then `(dy)/(dx)` at x = `1/2`, is equal to ______.
If log (x+y) = log (xy) + a then show that, `dy/dx= (-y^2)/(x^2)`
If log(x + y) = log(xy) + a then show that, `dy/dx = (-y^2)/x^2`
