3. Techniques of Differentiation

Find the derivative of the following functions.

y = e^-x sin x

Derivatives Find and simplify the derivative of the following functions.

h(x) = (x−1)(2x²-1) / (x³-1)

Derivatives Find and simplify the derivative of the following functions.

f(x) = x /x+1

9–61. Evaluate and simplify y'.

y = e^tan x (tan x−1)

Population growth Consider the following population functions.

e.Use a graphing utility to graph the population and its growth rate.

p(t) = 600 (t²+3/t²+9)

66–71. {Use of Tech} Estimating error Refer to Theorem 8.1 in the following exercises.

66. Let f(x) = cos(x²).

b. Calculate f''(x).

9–61. Evaluate and simplify y'.

y = (2x−3)x^3/2

Find an equation of the line tangent to the given curve at a.

y = (x + 5) / (x - 1); a = 3

Derivatives Find and simplify the derivative of the following functions.

f(t) = t⁵/³e^t

Evaluate and simplify y'.

y = 4u²+u / 8u+1

{Use of Tech} The Witch of Agnesi The graph of y = a³ / (x² + a²), where a is a constant, is called the witch of Agnesi (named after the 18th-century Italian mathematician Maria Agnesi).

Let a = 3 and find an equation of the line tangent to y = 27 / (x² + 9) at x = 2.

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

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

7–14. Find the derivative the following ways:

a. Using the Product Rule (Exercises 7–10) or the Quotient Rule (Exercises 11–14). Simplify your result.

f(w) = w³ -w / w

Evaluate and simplify y'.

y = (3x+5)¹⁰ √x²+5 / (x³+1)⁵⁰

Find the tangent line to the Witch of Agnesi (graphed here) at the point (2,1).