Derivative of ln|x| Differentiate ln x, for x > 0, and differentiate ln(−x), for x < 0, to conclude that d/dx (ln|x|) = 1/x

Finding Derivative Values

In Exercises 67–72, find the value of (f ∘ g)' at the given value of x.

f(u) = 1 − (1/u), u = g(x) = (1 / (1 − x)), x = −1

Given $z={\left(x-y\right)}^9$, where $x=s^{2}t$ and $y=s{t}^2$, use the chain rule to find the partial derivatives $\frac{\partial z}{\partial s}$ and $\frac{\partial z}{\partial t}$.

Given $z={\left(x-y\right)}^7$, where $x=s^{2}t$ and $y=s{t}^2$, use the chain rule to find the partial derivatives $\frac{\partial z}{\partial s}$ and $\frac{\partial z}{\partial t}$.

10–19. Derivatives Find the derivatives of the following functions.

f(x) = ln(3 sin² 4x)

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

59. y = √(t/(t+1))

In Exercises 59–86, find the derivative of y with respect to the given independent variable.

79. y = θ sin(log₇ θ)

In Exercises 59–86, find the derivative of y with respect to the given independent variable.

83. y = 3^(log₂ t)