6. Derivatives of Inverse, Exponential, & Logarithmic Functions

Calculate the derivative of the following functions. In some cases, it is useful to use the properties of logarithms to simplify the functions before computing f'(x).

f(x) = In 2x/(x² + 1)³

11–15. Identities Prove each identity using the definitions of the hyperbolic functions.

tanh(−x) = −tanh x

In Exercises 7–38, find the derivative of y with respect to x, t, or θ, as appropriate.

14. y = ln(2θ+2)

22–36. Derivatives Find the derivatives of the following functions.

f(t) = 2 tanh⁻¹ √t

Find the derivative of the given function.

$g\(\left\)(t\(\right\))=\(\log\)_5\(\left\)(7^{t^2+4}\(\right\))$

Find the derivative of the given function.

$f(x)=2x^3-1+\(\ln\) x$

Find the derivative of the following functions.

y = In x / (In x + 1)

Find the derivative of the given function.

$f(x)=2^x-5^x$

What is the domain of sech⁻¹ x? How is sech⁻¹ x defined in terms of the inverse hyperbolic cosine?

Find the derivative of the following functions.

y = x² In x

In Exercises 7–26, find the derivative of y with respect to x, t, or θ, as appropriate.

y = e^(θ)(sinθ + cosθ)

In Exercises 7–38, find the derivative of y with respect to x, t, or θ, as appropriate.

10. y = ln(t^(3/2))

Find the derivative of the given function.

$g(x)=x^4+4x+4^x$

Find the derivative of the given function.

$f(x)=(x^3+2x)\(\cdot\]\log\)_5x$

In Exercises 7–26, find the derivative of y with respect to x, t, or θ, as appropriate.

y = e^(5-7x)