6. Derivatives of Inverse, Exponential, & Logarithmic Functions

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

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^In x

Evaluate the following derivatives.

d/dx ((1/x)ˣ)

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

$y=\(\tan\)^{x}x$

In Exercises 1–24, find the derivative of y with respect to the appropriate variable.

13. y = (x+2)^(x+2)

Find f′(1) when f(x) = x^(1/x).

Evaluate and simplify y'.

y = (x²+1)³ / (x⁴+7)⁸(2x+1)⁷

9–61. Evaluate and simplify y'.

y = x^√x+1

Calculate the derivative of the following functions (i) using the fact that b^x = e^{xIn b} and (ii) using logarithmic differentiation. Verify that both answers are the same.

y = (x²+1)^x

In Exercises 115–126, use logarithmic differentiation or the method in Example 6 to find the derivative of y with respect to the given independent variable.

122. y = (ln x)^(ln x)

In Exercises 57–70, use logarithmic differentiation to find the derivative of y with respect to the given independent variable.

66. y = θsin(θ)/√(sec(θ))

In Exercises 115–126, use logarithmic differentiation or the method in Example 6 to find the derivative of y with respect to the given independent variable.

117. y = (√t)ᵗ

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

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