6. Derivatives of Inverse, Exponential, & Logarithmic Functions

Find the derivative of the given function.

$y=\tan \left(e^{-x^3}\right)$

Determine the intervals for which the function is concave up or concave down. State the inflection points.

$f(x)=4\(\ln\)(3x^2)$

Sketch the graphs of y = cosh x, y = sinh x, and y = tanh x (include asymptotes), and state whether each function is even, odd, or neither.

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

f(x) = sinh 4x

On what interval is the formula d/dx (tanh⁻¹ x) = 1/(1 - x²) valid?

Find the derivative of the following functions.

y = In |sin x|

16–18. Identities Use the given identity to prove the related identity.

Use the identity cosh 2x = cosh²x + sinh²x to prove the identities cosh²x = (cosh 2x + 1)/2 and sinh²x = (cosh 2x − 1)/2.

7–28. Derivatives Evaluate the following derivatives.

d/dx (ln³(3x² + 2))

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

18. y = t√(ln t)

7–28. Derivatives Evaluate the following derivatives.

d/dx ((2x)⁴ˣ)

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

33. y = ln(sec(lnθ))

The graph of y =x^{ln x} has one horizontal tangent line. Find an equation for it.

7–28. Derivatives Evaluate the following derivatives.

d/dx (sin (ln x))

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

29. y = ln(1/(x√(x+1)))

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

y = xe^x-e^x