3. Techniques of Differentiation

Evaluate and simplify y'.

y = 4u²+u / 8u+1

Evaluate and simplify y'.

y = (3x+5)¹⁰ √x²+5 / (x³+1)⁵⁰

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

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

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

c. d/dx ((f(x) / g(x)) |x=3

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

b. d/dx ((f(x) / g(x)) |x=

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

y = 4^-x sin x

9–61. Evaluate and simplify y'.

y = e^tan x (tan x−1)

9–61. Evaluate and simplify y'.

y = e^6x sin x

Find the derivative of the following functions.

y = e^-x sin x

Find the derivative of the following functions.

y = x sin x

Derivatives of sin^n x Calculate the following derivatives using the Product Rule.

c. d/dx (sin⁴ x)

Witch of Agnesi Let y(x²+4)=8 (see figure). <IMAGE>

c. Solve the equation y(x²+4)=8 for y to find an explicit expression for y and then calculate dy/dx.

Let ƒ(x) = (x - 3) (x + 3)²

a. Verify that ƒ'(x) = 3(x - 1) (x + 3) and ƒ"(x) = 6 (x + 1).

{Use of Tech} Beak length The length of the culmen (the upper ridge of a bird’s bill) of a t-week-old Indian spotted owlet is modeled by the function L(t)=11.94 / 1 + 4e^−1.65t, where L is measured in millimeters.

a. Find L′(1) and interpret the meaning of this value.

{Use of Tech} Beak length The length of the culmen (the upper ridge of a bird’s bill) of a t-week-old Indian spotted owlet is modeled by the function L(t)=11.94 / 1 + 4e^−1.65t, where L is measured in millimeters.

b. Use a graph of L′(t) to describe how the culmen grows over the first 5 weeks of life.