Advertisements
Advertisements
Question
If y = cos−1 [sin (4x)], find `("d"y)/("d"x)`
Solution
y = cos−1 [sin (4x)]
= `cos^-1 [cos(pi/2 - 4^x)]`
= y = `pi/2 - 4^x`
Differentiating w.r.t. x, we get
`("d"y)/("d"x) = "d"/("d"x)(pi/2 - 4^x)`
= 0 – 4x log 4
= – 4x log 4
APPEARS IN
RELATED QUESTIONS
If y = eax. cos bx, then prove that
`(d^2y)/(dx^2) - 2ady/dx + (a^2 + b^2)y` = 0
if `y = tan^2(log x^3)`, find `(dy)/(dx)`
Solve : `"dy"/"dx" = 1 - "xy" + "y" - "x"`
If y = log (cos ex) then find `"dy"/"dx".`
Find `"dy"/"dx"`If x3 + x2y + xy2 + y3 = 81
Find `dy/dx if x^2y^2 - tan^-1(sqrt(x^2 + y^2)) = cot^-1(sqrt(x^2 + y^2))`
Find `"dy"/"dx"` if ex+y = cos(x – y)
Find the second order derivatives of the following : xx
Find `"dy"/"dx"` if, y = `sqrt("x" + 1/"x")`
Find `"dy"/"dx"` if, y = (5x3 - 4x2 - 8x)9
Find `"dy"/"dx"` if, y = log(10x4 + 5x3 - 3x2 + 2)
Find `"dy"/"dx"` if, y = `"e"^(5"x"^2 - 2"x" + 4)`
Find `"dy"/"dx"` if, y = `"a"^((1 + log "x"))`
Choose the correct alternative.
If y = (5x3 - 4x2 - 8x)9, then `"dy"/"dx"` =
If y = 2x2 + 22 + a2, then `"dy"/"dx" = ?`
If `"x"^"m"*"y"^"n" = ("x + y")^("m + n")`, then `"dy"/"dx" = "______"/"x"`
State whether the following is True or False:
The derivative of polynomial is polynomial.
`d/dx(10^x) = x*10^(x - 1)`
Solve the following:
If y = (6x3 - 3x2 - 9x)10, find `"dy"/"dx"`
Find `"dy"/"dx"`, if y = xx.
Find `"dy"/"dx"`, if y = `2^("x"^"x")`.
Differentiate `"e"^("4x" + 5)` with respect to 104x.
If f'(4) = 5, f(4) = 3, g'(6) = 7 and R(x) = g[3 + f(x)] then R'(4) = ______
If x = cos−1(t), y = `sqrt(1 - "t"^2)` then `("d"y)/("d"x)` = ______
If x = f(t) and y = g(t) are differentiable functions of t so that y is a differentiable function of x and `(dx)/(dt)` ≠ 0 then `(dy)/(dx) = ((dy)/(dt))/((dx)/(d"))`.
Hence find `(dy)/(dx)` if x = sin t and y = cost
Choose the correct alternative:
If y = `root(3)((3x^2 + 8x - 6)^5`, then `("d"y)/("d"x)` = ?
If y = (5x3 – 4x2 – 8x)9, then `("d"y)/("d"x)` is ______
State whether the following statement is True or False:
If y = ex, then `("d"^2y)/("d"x^2)` = ex
Find `("d"y)/("d"x)`, if y = (6x3 – 3x2 – 9x)10
Find `("d"^2y)/("d"x^2)`, if y = `"e"^((2x + 1))`
y = (6x4 – 5x3 + 2x + 3)6, find `("d"y)/("d"x)`
Solution: Given,
y = (6x4 – 5x3 + 2x + 3)6
Let u = `[6x^4 - 5x^3 + square + 3]`
∴ y = `"u"^square`
∴ `("d"y)/"du"` = 6u6–1
∴ `("d"y)/"du"` = 6( )5
and `"du"/("d"x) = 24x^3 - 15(square) + 2`
By chain rule,
`("d"y)/("d"x) = ("d"y)/square xx square/("d"x)`
∴ `("d"y)/("d"x) = 6(6x^4 - 5x^3 + 2x + 3)^square xx (24x^3 - 15x^2 + square)`
If y = `2/(sqrt(a^2 - b^2))tan^-1[sqrt((a - b)/(a + b)) tan x/2], "then" (d^2y)/dx^2|_{x = pi/2}` = ______
Derivative of ex sin x w.r.t. e-x cos x is ______.
If y = (sin x2)2, then `("d"y)/("d"x)` is equal to ______.
`"d"/("d"x) [sin(1 - x^2)]^2` = ______.
If y = `(cos x)^((cosx)^((cosx))`, then `("d")/("d"x)` = ______.
If x = a sec3θ and y = a tan3θ, find `("d"y)/("d"x)` at θ = `pi/3`
If f(x) = |cos x|, find f'`((3pi)/4)`
If f(x) = |cos x – sinx|, find `"f'"(pi/6)`
If y = `sec^-1 ((sqrt(x) + 1)/(sqrt(x + 1))) + sin^-1((sqrt(x) - 1)/(sqrt(x) + 1))`, then `"dy"/"dx"` is equal to ______.
If y = log (cos ex), then `"dy"/"dx"` is:
y = `sec (tan sqrt(x))`
y = `cos sqrt(x)`
If ax2 + 2hxy + by2 = 0, then prove that `(d^2y)/(dx^2)` = 0.
If y = em sin–1 x and (1 – x2) = Ay2, then A is equal to ______.
Let f(x) = x | x | and g(x) = sin x
Statement I gof is differentiable at x = 0 and its derivative is continuous at that point.
Statement II gof is twice differentiable at x = 0.
If `d/dx` [f(x)] = ax+ b and f(0) = 0, then f(x) is equal to ______.
Solve the following:
If y = `root5 ((3x^2 + 8x + 5)^4 ,) "find" "dy"/ "dx"`
Find `dy/dx` if, y = `e^(5x^2 - 2x + 4)`
Find `dy/dx` if, y = `e^(5 x^2 - 2x + 4)`
Find `dy/dx` if ,
`x= e^(3t) , y = e^(4t+5)`
lf y = f(u) is a differentiable function of u and u = g(x) is a differentiable function of x, such that the composite function y = f[g(x)] is a differentiable function of x, then prove that:
`dy/dx = dy/(du) xx (du)/dx`
Hence, find `d/dx[log(x^5 + 4)]`.
If f(x) = `sqrt(7*g(x) - 3)`, g(3) = 4 and g'(3) = 5, find f'(3).
If y = `tan^-1((6x - 7)/(6 + 7x))`, then `dy/dx` = ______.
Find `dy/dx` if, y = `e^(5x^2 -2x + 4)`
Solve the following:
If y = `root5((3x^2 + 8x + 5)^4)`, find `dy/dx`
Find `dy/dx` if, y = `e^(5x^2-2x+4)`
Find `dy/dx` if, `y = e^(5x^2 - 2x +4)`
If `y=root5((3x^2+8x+5)^4)`, find `dy/dx`
Find `"dy"/"dx"` if, y = `"e"^(5"x"^2 - 2"x" + 4)`
Find `dy/dx` if, `y = e^(5x^2 - 2x+4)`
Solve the following.
If `y=root(5)((3x^2 + 8x + 5)^4)`, find `dy/dx`
Find `dy/(dx)` if, y = `e^(5x^2 - 2x + 4)`
If `y = (x + sqrt(a^2 + x^2))^m`, prove that `(a^2 + x^2)(d^2y)/(dx^2) + xdy/dx - m^2y = 0`
Find `dy/dx` if, `y = e^(5x^2 - 2x + 4)`.
