6. Derivatives of Inverse, Exponential, & Logarithmic Functions

Identify the local minimum and maximum values of the given function, if any.

$f(t)=t^2\(\ln\) t$, $t>0$

15–48. Derivatives Find the derivative of the following functions.

P = 40/1+2^-t

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

35. y = ln((x²+1)^5/√(1-x))

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

16. y = (ln x)³

Express sinh⁻¹ x in terms of logarithms.

Find d/dx(ln√x²+1).

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(sec⁴x tan² x)

Find the derivative of the following functions.

y = In √x⁴+x²

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

f(x) = x² cosh² 3x

15–48. Derivatives Find the derivative of the following functions.

y = 10^x(In 10^x-1)

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).

y = (cos x) In cos²x

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

y = ln(e^(θ)/(1+e^θ))

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

5. y = ln(sin²θ)

7–28. Derivatives Evaluate the following derivatives.

d/dx (ln (cos² x))

A calculator has a built-in sinh⁻¹ x function, but no csch⁻¹ x function. How do you evaluate csch⁻¹ 5 on such a calculator?