Advertisements
Advertisements
प्रश्न
Differentiate the function with respect to x.
`x^x - 2^(sin x)`
उत्तर
Let, y = xx - 2sin x
Again, let u = xx, v = 2sin x
y = u - v
Taking logarithm of both sides of u = xx,
log u = log xx = x log x
Differentiating both sides with respect to x,
`1/u (du)/dx = x d/dx log x + log x d/dx (x)`
`=> 1/u (du)/dx = x * 1/x + log x xx 1/u (du)/dx = 1 + log x` ...(1)
`therefore (du)/dx = u (1 + log x) = x^x (1 + log x)` ...(2)
Now, from `v = 2^(sin x)`
`(dv)/dx= 2^ (sin x) log 2 d/dx (sin x)`
`= 2^(sin x) log 2 cos x` ...(3)
From equation (1), y = u – v
`therefore dy/dx = (du)/dx - (dv)/dx`
Putting the values of `(du)/dx` from equation (2) and `(dv)/dx` from (3),
`dy/dx = x^x (1 + log x) - 2^(sin x) (cos x. log 2)`
APPEARS IN
संबंधित प्रश्न
Differentiate the following function with respect to x: `(log x)^x+x^(logx)`
Differentiate the function with respect to x.
`sqrt(((x-1)(x-2))/((x-3)(x-4)(x-5)))`
Differentiate the function with respect to x.
`(log x)^(cos x)`
Differentiate the function with respect to x.
(log x)x + xlog 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
Find `dy/dx` for the function given in the question:
`xy = e^((x – y))`
If cos y = x cos (a + y), with cos a ≠ ± 1, prove that `dy/dx = cos^2(a+y)/(sin a)`
if `x^m y^n = (x + y)^(m + n)`, prove that `(d^2y)/(dx^2)= 0`
If `y = sin^-1 x + cos^-1 x , "find" dy/dx`
Find `(dy)/(dx) , if y = sin ^(-1) [2^(x +1 )/(1+4^x)]`
Find `dy/dx` if y = xx + 5x
Differentiate
log (1 + x2) w.r.t. tan-1 (x)
Find `"dy"/"dx"` , if `"y" = "x"^("e"^"x")`
If `log_5((x^4 + y^4)/(x^4 - y^4)) = 2, "show that""dy"/"dx" = (12x^3)/(13y^3)`.
If xy = ex–y, then show that `"dy"/"dx" = logx/(1 + logx)^2`.
If ey = yx, then show that `"dy"/"dx" = (logy)^2/(log y - 1)`.
If x = `asqrt(secθ - tanθ), y = asqrt(secθ + tanθ), "then show that" "dy"/"dx" = -y/x`.
If x = log(1 + t2), y = t – tan–1t,show that `"dy"/"dx" = sqrt(e^x - 1)/(2)`.
Differentiate 3x w.r.t. logx3.
Find the second order derivatives of the following : x3.logx
Find the nth derivative of the following : log (ax + b)
If y = A cos (log x) + B sin (log x), show that x2y2 + xy1 + y = 0.
If y = `log[4^(2x)((x^2 + 5)/sqrt(2x^3 - 4))^(3/2)]`, find `("d"y)/("d"x)`
If y = 5x. x5. xx. 55 , find `("d"y)/("d"x)`
The rate at which the metal cools in moving air is proportional to the difference of temperatures between the metal and air. If the air temperature is 290 K and the metal temperature drops from 370 K to 330 K in 1 O min, then the time required to drop the temperature upto 295 K.
If y = tan-1 `((1 - cos 3x)/(sin 3x))`, then `"dy"/"dx"` = ______.
Derivative of `log_6`x with respect 6x to is ______
If y `= "e"^(3"x" + 7), "then the value" |("dy")/("dx")|_("x" = 0)` is ____________.
If `log_10 ((x^2 - y^2)/(x^2 + y^2))` = 2, then `dy/dx` is equal to ______.
The derivative of log x with respect to `1/x` is ______.
Find the derivative of `y = log x + 1/x` with respect to x.
If xy = yx, then find `dy/dx`
