3. Techniques of Differentiation

In Exercises 9–18, write the function in the form y = f(u) and u = g(x). Then find dy/dx as a function of x.

y = (2x + 1)⁵

In Exercises 9–18, write the function in the form y = f(u) and u = g(x). Then find dy/dx as a function of x.

y = (4 − 3x)⁹

Find the derivatives of the functions in Exercises 19–40.

p = √(3 − t)

Find the derivatives of the functions in Exercises 19–40.

s = (4 / 3π)sin(3t) + (4 / 5π)cos(5t)

Find the derivatives of the functions in Exercises 19–40.

y = (5 − 2x)⁻³ + (1 / 8)(2 / x + 1)⁴

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

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

f(x) = ln(3 sin² 4x)

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

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)

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

65. y = (cos θ)^(√2)

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

67. y = 7^(sec θ) ln 7"

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

69. y = 2^(sin 3t)

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

71. y = log₂(5θ)