3. Techniques of Differentiation

Find the derivatives of the functions in Exercises 1–42.

𝔂 = 2 tan² x - sec² x

In Exercises 41–58, find dy/dt.

y = (1/6)(1 + cos²(7t))³

{Use of Tech} Cell population The population of a culture of cells after t days is approximated by the function P(t)=1600 / 1 + 7e^−0.02t, for t≥0.

a. Graph the population function.  

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

s = (4 / 3π)sin(3t) + (4 / 5π)cos(5t)

27–76. Calculate the derivative of the following functions.

y = √f(x), where f is differentiable and nonnegative at x.

In Exercises 41–58, find dy/dt.

y = tan²(sin³(t))

109-112 {Use of Tech} Calculating limits The following limits are the derivatives of a composite function g at a point a.

b. Use the Chain Rule to find each limit. Verify your answer by using a calculator.

$\underset{x\to 0}{\lim }\frac{\sqrt{4+\sin \left(x\right)}-2}{x}$

Calculate the derivative of the following functions.

y = cos⁴ θ + sin⁴ θ

5–24. For each of the following composite functions, find an inner function u=g(x) and an outer function y=f(u) such that y=f(g(x)). Then calculate dy/dx.

y = (5x²+11x)^4/3

Second derivatives Find d²y/dx²for the following functions.

y = √x²+2

Composition containing sin x Suppose f is differentiable for all real numbers with f(0)=−3,f(1)=3,f′(0)=3, and f′(1)=5. Let g(x)=sin(πf(x)). Evaluate the following expressions.

b. g'(1)

Vibrations of a spring Suppose an object of mass m is attached to the end of a spring hanging from the ceiling. The mass is at its equilibrium position $y=0$ when the mass hangs at rest. Suppose you push the mass to a position $y_0$ units above its equilibrium position and release it. As the mass oscillates up and down (neglecting any friction in the system), the position y of the mass after t seconds is $y=y_0\cos \left(t\sqrt{\frac{k}{m}}\right)$, where $k>0$ is a constant measuring the stiffness of the spring (the larger the value of $k$, the stiffer the spring) and $y$ is positive in the upward direction.

Use equation (4) to answer the following questions.

c. How would the velocity be affected if the experiment were repeated with a spring having four times the stiffness ($k$ is increased by a factor of $4$)?

Find the derivatives of the functions in Exercises 1–42.

𝔂 = x² sin² (2x²)

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

y = 5^3t

Finding Derivative Values

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

f(u) = u + 1/cos²u, u = g(x) = πx, x = 1/4