6. Derivatives of Inverse, Exponential, & Logarithmic Functions

Use logarithmic differentiation to find the derivative of the given function.

$y=(\sqrt{x+1})^{x}$

Use logarithmic differentiation to find the derivative of the given function.

$y=\sqrt{\left(\frac{x+2}{x}\right)^{\frac23}\left(x^2-4\right)}$

Use logarithmic differentiation to find the derivative of the given function.

$y=\tan^{x}x$

49–55. Derivatives of tower functions (or g^h) Find the derivative of each function and evaluate the derivative at the given value of a.

g (x) = x^ In x; a = e

Find f′(1) when f(x) = x^(1/x).

49–55. Derivatives of tower functions (or g^h) Find the derivative of each function and evaluate the derivative at the given value of a.

h (x) = x^√x; a = 4

49–55. Derivatives of tower functions (or g^h) Find the derivative of each function and evaluate the derivative at the given value of a.

f (x) = (sin x)^In x; a = π/2

Evaluate and simplify y'.

y = (x²+1)³ / (x⁴+7)⁸(2x+1)⁷

49–55. Derivatives of tower functions (or g^h) Find the derivative of each function and evaluate the derivative at the given value of a.

f (x) = (4 sin x+2)^cos x; a = π

9–61. Evaluate and simplify y'.

y = x^√x+1

75–86. Logarithmic differentiation Use logarithmic differentiation to evaluate f'(x).

f(x) = x^10x

75–86. Logarithmic differentiation Use logarithmic differentiation to evaluate f'(x).

f(x) = (x+1)¹⁰ / (2x-4)⁸

75–86. Logarithmic differentiation Use logarithmic differentiation to evaluate f'(x).

f(x) = x^In x

75–86. Logarithmic differentiation Use logarithmic differentiation to evaluate f'(x).

f(x) = tan¹⁰x / (5x+3)⁶

75–86. Logarithmic differentiation Use logarithmic differentiation to evaluate f'(x).

f(x) = (x+1)^3/2(x-4)^5/2 / (5x+3)^2/3