3. Techniques of Differentiation

Finding derivatives from a table Find the values of the following derivatives using the table. <IMAGE>

d. d/dx (f(x)³) |x=5

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

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

{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.  

{Use of Tech} Tangent lines Determine equations of the lines tangent to the graph of y= x√5−x² at the points (1, 2) and (−2,−2). Graph the function and the tangent lines.

9–61. Evaluate and simplify y'.

y = (v / v+1)^4/3

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.

$\underset{x\to 2}{\lim }\frac{{(x^2-3)}^5-1}{x-2}$

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.

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

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.

$\underset{h\to 0}{\lim }\frac{\frac{1}{3{({(1+h)}^5+7)}^{10}}-\frac{1}{3{\left(8\right)}^{10}}}{h}$

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

d/dx (f(f(x))) |x=4

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 $\frac{d^2}{d{x}^2}\left(\sin (3x^4+5x^2+2)\right)$.