Advertisements
Advertisements
प्रश्न
If y = `log[4^(2x)((x^2 + 5)/sqrt(2x^3 - 4))^(3/2)]`, find `("d"y)/("d"x)`
Advertisements
उत्तर
y = `log[4^(2x)((x^2 + 5)/sqrt(2x^3 - 4))^(3/2)]`
= `log4^(2x) + log((x^2 + 5)/sqrt(2x^3 - 4))^(3/2)`
= `2x log4 + 3/2[log(x^2 + 5)/(sqrt(2x^3 - 4))]`
= `2x log4 + 3/2[log(x^2 + 5) - logsqrt(2^3 - 4)]`
= `2x log4 + 3/2[log(x^2 + 5) - 3/4log(2x^3 - 4)]`
Differentiating w. r. t. x, we get
`("d"y)/("d"x) = "d"/("dx)[2xlog 4 + 3/2 log(x^2 + 5) - 3/4log(2x^3 - 4)]`
= `2log4*"d"/("d"x)(x) + 3/2*"d"/("d"x) [log(x^2 + 5)] - 3/4*"d"/("d"x) [log(2x^3 - 4)]`
= `2log4*1 + 3/*1/(x^2 + 5)*"d"/("d"x) (x^2 + 5) - 3/4*1/(2x^3 - 4)*"d"/("d"x)(2x^3 - 4)`
= `2log4 + 3/2*1/(x^2 + 5)*2x - 3/4*1/(2x^3 - 4)*6x^2`
∴ `("d"y)/("d"x) = 2log4 + (3x)/(x^2 + 5) - (9x^2)/(2(2x^3 - 4)`
APPEARS IN
संबंधित प्रश्न
Differentiate the following function with respect to x: `(log x)^x+x^(logx)`
If `y=log[x+sqrt(x^2+a^2)]` show that `(x^2+a^2)(d^2y)/(dx^2)+xdy/dx=0`
Differentiate the function with respect to x.
cos x . cos 2x . cos 3x
Differentiate the function with respect to x.
xx − 2sin x
Differentiate the function with respect to x.
`(sin x)^x + sin^(-1) sqrtx`
Differentiate the function with respect to x.
xsin x + (sin x)cos x
Differentiate the function with respect to x.
`x^(xcosx) + (x^2 + 1)/(x^2 -1)`
Differentiate the function with respect to x.
`(x cos x)^x + (x sin x)^(1/x)`
Find `bb(dy/dx)` for the given function:
xy + yx = 1
Find `bb(dy/dx)` for the given function:
yx = xy
If u, v and w are functions of x, then show that `d/dx(u.v.w) = (du)/dx v.w + u. (dv)/dx.w + u.v. (dw)/dx` in two ways-first by repeated application of product rule, second by logarithmic differentiation.
Differentiate the function with respect to x:
xx + xa + ax + aa, for some fixed a > 0 and x > 0
Find `(dy)/(dx) , if y = sin ^(-1) [2^(x +1 )/(1+4^x)]`
Evaluate
`int 1/(16 - 9x^2) dx`
Solve the following differential equation: (3xy + y2) dx + (x2 + xy) dy = 0
If log (x + y) = log(xy) + p, where p is a constant, then prove that `"dy"/"dx" = (-y^2)/(x^2)`.
If xy = ex–y, then show that `"dy"/"dx" = logx/(1 + logx)^2`.
If x = `asqrt(secθ - tanθ), y = asqrt(secθ + tanθ), "then show that" "dy"/"dx" = -y/x`.
If x = esin3t, y = ecos3t, then show that `dy/dx = -(ylogx)/(xlogy)`.
If x = 2cos4(t + 3), y = 3sin4(t + 3), show that `"dy"/"dx" = -sqrt((3y)/(2x)`.
If x = log(1 + t2), y = t – tan–1t,show that `"dy"/"dx" = sqrt(e^x - 1)/(2)`.
If y = `log(x + sqrt(x^2 + a^2))^m`, show that `(x^2 + a^2)(d^2y)/(dx^2) + x "d"/"dx"` = 0.
If y = log (log 2x), show that xy2 + y1 (1 + xy1) = 0.
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 log5 `((x^4 + "y"^4)/(x^4 - "y"^4))` = 2, show that `("dy")/("d"x) = (12x^3)/(13"y"^2)`
If y = `{f(x)}^{phi(x)}`, then `dy/dx` is ______
If y = tan-1 `((1 - cos 3x)/(sin 3x))`, then `"dy"/"dx"` = ______.
`d/dx(x^{sinx})` = ______
If `("f"(x))/(log (sec x)) "dx"` = log(log sec x) + c, then f(x) = ______.
If y = `("e"^"2x" sin x)/(x cos x), "then" "dy"/"dx" = ?`
`8^x/x^8`
If `"y" = "e"^(1/2log (1 + "tan"^2"x")), "then" "dy"/"dx"` is equal to ____________.
If `f(x) = log [e^x ((3 - x)/(3 + x))^(1/3)]`, then `f^'(1)` is equal to
Derivative of log (sec θ + tan θ) with respect to sec θ at θ = `π/4` is ______.
Find `dy/dx`, if y = (sin x)tan x – xlog x.
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 ______.
Find the derivative of `y = log x + 1/x` with respect to x.
