6. Derivatives of Inverse, Exponential, & Logarithmic Functions

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

f(x) = x⁸cos³ x / √x-1

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

f(x) = (1+ 1/x)^x

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

Evaluate the following derivatives.

d/dx (x^{x¹⁰})

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

70. y = ∛(x(x+1)(x-2)/(x²+1)(2x+3))

In Exercises 25–30, use logarithmic differentiation to find the derivative of y with respect to the appropriate variable.

29. y = (sin θ)^√θ

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

$y=\(\sqrt{\left(\frac{x+2}{x}\]\right\))^{\(\frac\)23}\(\left\)(x^2-4\(\right\))}$

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

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

119. y = (sin x)ˣ

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 = (4x+1)^{In x}

In Exercises 25–30, use logarithmic differentiation to find the derivative of y with respect to the appropriate variable.

25. y = 2(x² + 1)/√(cos 2x)

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.

120. y = x^(sin 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.

124. x^(sin y) = ln y

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

61. y = √(θ + 3) sin θ