Advertisements
Advertisements
Question
Differentiate the following w.r.t. x:
`tan^-1(x/(1 + 6x^2)) + cot^-1((1 - 10x^2)/(7x))`
Advertisements
Solution
Let y = `tan^-1(x/(1 + 6x^2)) + cot^-1((1 - 10x^2)/(7x))`
= `tan^-1(x/(1 + 6x^2)) + cot^-1((1 - 10x^2)/(7x)) ...[∵ cot^-1x = tan^-1(1/x)]`
= `tan^-1(x/(1 + 6x^2)) + tan^-1((7x)/(1 - 10x^2)) ...[∵ cot^-1x = tan^-1(1/x)]`
= `tan^-1[(3x - 2x)/(1 + (3)(2x))] + tan^-1[(5x + 2x)/(1 - (5x)(2x))]`
= tan–13x – tan–12x + tan–15x + tan–12x
= tan–13x + tan–15x
∴ `"dy"/"dx" = "d"/"dx"[tan^-1 3x + tan^-1 5x]`
= `"d"/"dx"(tan^-1 3x) + "d"/"dx"(tan^-1 5x)`
= `(1)/(1 + (3x)^2)."d"/"dx"(3x) + (1)/(1 + (5x)^2)."d"/"dx"(5x)`
= `(1)/(1 + 9x^2) xx 3 xx 1 + (1)/(1 + 25x^2) xx 5 xx 1`
= `(3)/(1 + 9x^2) + (5)/(1 + 25x^2`
APPEARS IN
RELATED QUESTIONS
If xpyq = (x + y)p+q then Prove that `dy/dx = y/x`
if `x^y + y^x = a^b`then Find `dy/dx`
Find the derivative of the function f defined by f (x) = mx + c at x = 0.
Write the value of the derivative of f (x) = |x − 1| + |x − 3| at x = 2.
Find `dy/dx if x^3 + y^2 + xy = 7`
Differentiate e4x + 5 w.r..t.e3x
Find `(dy)/(dx) if y = cos^-1 (√x)`
Discuss extreme values of the function f(x) = x.logx
If ex + ey = ex+y, then show that `"dy"/"dx" = -e^(y - x)`.
Find `"dy"/"dx"` if x = a cot θ, y = b cosec θ
Find `"dy"/"dx"`, if : x = `(t + 1/t)^a, y = a^(t+1/t)`, where a > 0, a ≠ 1, t ≠ 0.
Differentiate `tan^-1((x)/(sqrt(1 - x^2))) w.r.t. sec^-1((1)/(2x^2 - 1))`.
Differentiate `tan^-1((cosx)/(1 + sinx)) w.r.t. sec^-1 x.`
Differentiate `tan^-1((sqrt(1 + x^2) - 1)/(x)) w.r.t tan^-1((2xsqrt(1 - x^2))/(1 - 2x^2))`.
If `sec^-1((7x^3 - 5y^3)/(7^3 + 5y^3)) = "m", "show" (d^2y)/(dx^2)` = 0.
Find the nth derivative of the following : (ax + b)m
Find the nth derivative of the following : eax+b
Find the nth derivative of the following : cos (3 – 2x)
Choose the correct option from the given alternatives :
Let `f(1) = 3, f'(1) = -(1)/(3), g(1) = -4 and g'(1) =-(8)/(3).` The derivative of `sqrt([f(x)]^2 + [g(x)]^2` w.r.t. x at x = 1 is
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 `xsqrt(y + 1) + ysqrt(x + 1) = 0 and x ≠ y, "then" "dy"/"dx"` = ........
Suppose that the functions f and g and their derivatives with respect to x have the following values at x = 0 and x = 1:
| x | f(x) | g(x) | f')x) | g'(x) |
| 0 | 1 | 5 | `(1)/(3)` | |
| 1 | 3 | – 4 | `-(1)/(3)` | `-(8)/(3)` |
(i) The derivative of f[g(x)] w.r.t. x at x = 0 is ......
(ii) The derivative of g[f(x)] w.r.t. x at x = 0 is ......
(iii) The value of `["d"/"dx"[x^(10) + f(x)]^(-2)]_(x = 1_` is ........
(iv) The derivative of f[(x + g(x))] w.r.t. x at x = 0 is ...
Differentiate the following w.r.t. x : `cos^-1((sqrt(1 + x) - sqrt(1 - x))/2)`
Differentiate log `[(sqrt(1 + x^2) + x)/(sqrt(1 + x^2 - x)]]` w.r.t. cos (log x).
If y2 = a2cos2x + b2sin2x, show that `y + (d^2y)/(dx^2) = (a^2b^2)/y^3`
If y = Aemx + Benx, show that y2 – (m + n)y1 + mny = 0.
Find `"dy"/"dx"` if, x3 + y3 + 4x3y = 0
Find `"dy"/"dx"` if, x3 + x2y + xy2 + y3 = 81
Find `"dy"/"dx"` if, `"x"^"y" = "e"^("x - y")`
Choose the correct alternative.
If ax2 + 2hxy + by2 = 0 then `"dy"/"dx" = ?`
Choose the correct alternative.
If `"x"^4."y"^5 = ("x + y")^("m + 1")` then `"dy"/"dx" = "y"/"x"` then m = ?
State whether the following is True or False:
The derivative of `"x"^"m"*"y"^"n" = ("x + y")^("m + n")` is `"x"/"y"`
If x2 + y2 = 1, then `(d^2x)/(dy^2)` = ______.
If x2 + y2 = t + `1/"t"` and x4 + y4 = t2 + `1/"t"^2` then `("d"y)/("d"x)` = ______
If x = a t4 y = 2a t2 then `("d"y)/("d"x)` = ______
If y = `sqrt(tansqrt(x)`, find `("d"y)/("d"x)`.
If `sqrt(x) + sqrt(y) = sqrt("a")`, then `("d"y)/("d"x)` is ______
`(dy)/(dx)` of `2x + 3y = sin x` is:-
`(dy)/(dx)` of `xy + y^2 = tan x + y` is
y = `e^(x3)`
Find `(dy)/(dx)` if x + sin(x + y) = y – cos(x – y)
If log(x+y) = log(xy) + a then show that, `dy/dx = (-y^2)/x^2`
If y = `(x + sqrt(x^2 - 1))^m`, show that `(x^2 - 1)(d^2y)/(dx^2) + xdy/dx` = m2y
Find `dy / dx` if, x = `e^(3t), y = e^sqrt t`
Solve the following.
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`
Find `dy/dx` if, `x = e^(3t), y = e^(sqrtt)`
