3. Techniques of Differentiation

Use the given graphs of f and g to find each derivative. <IMAGE>

d/dx (g(f(x))) |x=1

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)

Composition containing sin x Suppose f is differentiable on [−2,2] with f′(0)=3 and f′(1)=5. Let g(x)=f(sin x). Evaluate the following expressions.

c. g'(π)

Tangent lines Assume f is a differentiable function whose graph passes through the point (1, 4). Suppose g(x)=f(x²) and the line tangent to the graph of f at (1, 4) is y=3x+1. Find each of the following.

a. g(1)

{Use of Tech} Hours of daylight The number of hours of daylight at any point on Earth fluctuates throughout the year. In the Northern Hemisphere, the shortest day is on the winter solstice and the longest day is on the summer solstice. At 40° north latitude, the length of a day is approximated by D(t) = 12−3 cos (2π(t+10) / 365), where D is measured in hours and 0≤t≤365 is measured in days, with t=0 corresponding to January 1.

b. Find the rate at which the daylight function changes.

The Chain Rule for second derivatives

b. Use the formula in part (a) to calculate $\frac{d^2}{d{x}^2}\left(\sin (3x^4+5x^2+2)\right)$.

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