6. Derivatives of Inverse, Exponential, & Logarithmic Functions

Find the derivative of the given function.

$g(t)=2t+\(\log\)_5t$

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

27. y = θ(sin(lnθ) + cos(lnθ))

In Exercises 13–24, find the derivative of y with respect to the appropriate variable.

23. y = (x²+1)sech(ln x)

(Hint: Before differentiating, express in terms of exponentials and simplify.)

Find the derivative of the following functions.

y = In |x²-1|

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

f(x) = x sinh⁻¹ x − √(x² + 1)

In Exercises 13–24, find the derivative of y with respect to the appropriate variable.

13. y = 6sinh(x/3)

Find the derivative of the following functions.

y = x In x - x

In Exercises 13–24, find the derivative of y with respect to the appropriate variable.

21. y = ln(cosh v) - 1/2 tanh²v

Determine whether the following statements are true and give an explanation or counterexample.

d/dx((√2)^x) = x(√2)^{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).

f(x) = In (2x - 1)(x + 2)³ / (1 - 4x)²

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

s(t) = cos 2^t

Find d/dx(ln(x/x²+1)) without using the Quotient Rule.

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

f(x) = cosh²x

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

y = (x^2 - 2x + 2)e^(x)

Find the derivative of the given function.

$f(x)=8\(\log\)_2x$