3. Techniques of Differentiation

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

67. Let f(x) = √(x³ + 1).

b. Calculate f''(x).

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.

y = x² - 2ax +a² / x-a, where a is a constant

First and second derivatives Find f′(x),f′′(x).

f(x) = x/x+2

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.

y = x² - a² / x-a, where a is a constant

Find the slope of the graph of f(x) = 2 + xe^x at the point (0, 2).

Suppose that functions ƒ(x) and g(x) and their first derivatives have the following values at x = 0 and x = 1.

x ƒ(x) g(x) ƒ'(x) g'(x)

0 1 1 -3 1/2

1 3 5 1/2 -4

Find the first derivatives of the following combinations at the given value of x.

b. ƒ(x)g²(x), x = 0

Derivatives Find and simplify the derivative of the following functions.

g(x) = x⁴/³-1 / x⁴/³+1

Find the derivative the following ways:

Using the Product Rule or the Quotient Rule. Simplify your result.

h(z) = (z³ + 4z² + z)(z - 1)

Let ƒ(x) = (x - 3) (x + 3)²

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

Product Rule for three functions Assume f, g, and h are differentiable at x.

b. Use the formula in (a) to find d/dx(e^x(x−1)(x+3))

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

𝔂 = 1 x² csc 2

2 x

Find the derivative of each function.

$y=(3x+5)^2$

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

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

Find the derivative of each function.

$f\left(t\right)=2t(t^{-3}+t^{2/3})$

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.