Derivative Calculations


In Exercises 1–8, given y = f(u) and u = g(x), find dy/dx = f'(g(x)) g'(x).


y = sin u, u = x − cos x

Falling meteorite The velocity of a heavy meteorite entering Earth’s atmosphere is inversely proportional to √s when it is s km from Earth’s center. Show that the meteorite’s acceleration is inversely proportional to s².

Derivative Calculations

In Exercises 1–8, given y = f(u) and u = g(x), find dy/dx = f'(g(x)) g'(x).

y = sin u, u = 3x + 1

Derivative Calculations

In Exercises 1–8, given y = f(u) and u = g(x), find dy/dx = f'(g(x)) g'(x).

y = cos u, u = −x/3

Derivative Calculations

In Exercises 1–8, given y = f(u) and u = g(x), find dy/dx = f'(g(x)) g'(x).

y = √u, u = sin x

Temperature and the period of a pendulum For oscillations of small amplitude (short swings), we may safely model the relationship between the period T and the length L of a simple pendulum with the equation:

T = 2π√(L/g),

where g is the constant acceleration of gravity at the pendulum’s location. If we measure g in centimeters per second squared, we measure L in centimeters and T in seconds. If the pendulum is made of metal, its length will vary with temperature, either increasing or decreasing at a rate that is roughly proportional to L. In symbols, with u being temperature and k the proportionality constant,

dL/du = kL.

Assuming this to be the case, show that the rate at which the period changes with respect to temperature is kT/2.

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)