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+x²)^sin x

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

f(x) = (1+ 1/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 = (x²+1)^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}

Evaluate the following derivatives.

d/dx ((1/x)ˣ)

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.

61. y = √(θ + 3) sin θ

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

64. y = 1/(t(t+1)(t+2))

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

115. y = (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.

117. y = (√t)ᵗ

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

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)