Advertisements
Advertisements
Question
Find `(dy)/(dx) , if y = sin ^(-1) [2^(x +1 )/(1+4^x)]`
Advertisements
Solution
`y = sin^(-1) [(2.2^x)/(1 +(2^x)^2)]`
put 2x = tan θ
`∴ y = sin^(-1) [(2 tan theta ) /(1 + tan^2 theta)]`
= sin-1 [ sin 2θ ]
= 2θ
y = 2 tan-1 ( 2x )
Differentiating wrt x,
`(dy)/(dx) = 2/(1 +(2^x) )xx d/(dx) (2^x)`
`= 2/(1 + (2^x)^2) xx 2^x log 2 = (2 ^ (x+ 1))/(1 + 4^x) log 2 =" sin y log" 2`
APPEARS IN
RELATED QUESTIONS
Differentiate the following function with respect to x: `(log x)^x+x^(logx)`
Differentiate the function with respect to x.
(log x)x + xlog x
Differentiate the function with respect to x.
`x^(xcosx) + (x^2 + 1)/(x^2 -1)`
Find `bb(dy/dx)` for the given function:
yx = xy
If cos y = x cos (a + y), with cos a ≠ ± 1, prove that `dy/dx = cos^2(a+y)/(sin a)`.
If x = a (cos t + t sin t) and y = a (sin t – t cos t), find `(d^2y)/dx^2`.
If y = `e^(acos^(-1)x)`, −1 ≤ x ≤ 1, show that `(1- x^2) (d^2y)/(dx^2) -x dy/dx - a^2y = 0`.
Differentiate
log (1 + x2) w.r.t. tan-1 (x)
Differentiate : log (1 + x2) w.r.t. cot-1 x.
If `"x"^(5/3) . "y"^(2/3) = ("x + y")^(7/3)` , the show that `"dy"/"dx" = "y"/"x"`
If log (x + y) = log(xy) + p, where p is a constant, then prove that `"dy"/"dx" = (-y^2)/(x^2)`.
If `log_5((x^4 + y^4)/(x^4 - y^4)) = 2, "show that""dy"/"dx" = (12x^3)/(13y^3)`.
If x = 2cos4(t + 3), y = 3sin4(t + 3), show that `"dy"/"dx" = -sqrt((3y)/(2x)`.
Differentiate 3x w.r.t. logx3.
Find the nth derivative of the following : log (2x + 3)
If y = A cos (log x) + B sin (log x), show that x2y2 + xy1 + y = 0.
If f(x) = logx (log x) then f'(e) is ______
If y = log [cos(x5)] then find `("d"y)/("d"x)`
If y = `(sin x)^sin x` , then `"dy"/"dx"` = ?
If y = `{f(x)}^{phi(x)}`, then `dy/dx` is ______
If `("f"(x))/(log (sec x)) "dx"` = log(log sec x) + c, then f(x) = ______.
`log [log(logx^5)]`
If `"f" ("x") = sqrt (1 + "cos"^2 ("x"^2)), "then the value of f'" (sqrtpi/2)` is ____________.
Given f(x) = `log((1 + x)/(1 - x))` and g(x) = `(3x + x^3)/(1 + 3x^2)`, then fog(x) equals
Derivative of log (sec θ + tan θ) with respect to sec θ at θ = `π/4` is ______.
If y = `log(x + sqrt(x^2 + 4))`, show that `dy/dx = 1/sqrt(x^2 + 4)`
The derivative of log x with respect to `1/x` is ______.
If xy = yx, then find `dy/dx`
If \[y=x^x+x^{\frac{1}{x}}\] then \[\frac{\mathrm{d}y}{\mathrm{d}x}\] is equal to
