3. Techniques of Differentiation

Deriving trigonometric identities

a. Differentiate both sides of the identity cos 2t = cos² t−sin² t to prove that sin 2 t= 2 sin t cos t.

Deriving trigonometric identities

b. Verify that you obtain the same identity for sin2t as in part (a) if you differentiate the identity cos 2t = 2 cos² t−1.

Deriving trigonometric identities

c. Differentiate both sides of the identity sin 2t = 2 sin t cost to prove that cos 2t = cos²t−sin²t.

{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.

e. Graph the growth rate. When is it a maximum and what is the population at the time that the growth rate is a maximum? 

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

y = In (x³+1)^π

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

y = 5^3t

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

y = 10^In 2x

9–61. Evaluate and simplify y'.

y = e^2θ

9–61. Evaluate and simplify y'.

y = e^sin x+2x+1

9–61. Evaluate and simplify y'.

y = e^sin (cosx)

{Use of Tech} Tangent line Find the equation of the line tangent to y=2^sin x at x=π/2. Graph the function and the tangent line.

9–61. Evaluate and simplify y'.

y = 10^sin x+sin¹⁰x

Applying the Chain Rule Use the data in Tables 3.4 and 3.5 of Example 4 to estimate the rate of change in pressure with respect to time experienced by the runner when she is at an altitude of 13,330 ft. Make use of a forward difference quotient when estimating the required derivatives.

Find the value of dy/dt at t = 0 if y = 3 sin 2x and x = t² + π.

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

𝔂 = 3 .

(5x² + sin 2x)³/²