3. Techniques of Differentiation

Find the derivative the following ways:

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

f(x) = (x - 1)(3x + 4)

Find the derivative the following ways:

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

h(z) = (z³ + 4z² + z)(z - 1)

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

f(x) = x/x+2

Derivatives from a table Use the following table to find the given derivatives. <IMAGE>

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

Derivatives from graphs Use the figure to find the following derivatives. <IMAGE>

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

Derivatives from graphs Use the figure to find the following derivatives. <IMAGE>

d/dx (xg(x)) | x=2

The line tangent to the curve y=h(x) at x=4 is y = −3x+14. Find an equation of the line tangent to the following curves at x=4.

y = (x²-3x)h(x)

Suppose the line tangent to the graph of f at x=2 is y=4x+1 and suppose y=3x−2 is the line tangent to the graph of g at x=2. Find an equation of the line tangent to the following curves at x=2.

y = f(x)g(x)

Given that p(x) = (5e^x+10x⁵+20x³+100x²+5x+20) ⋅ (10x⁵+40x³+20x²+4x+10), find p′(0) without computing p′(x).

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

a. Use the Product Rule (twice) to find a formula for d/dx (f(x)g(x)h(x)).

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

Derivatives Find and simplify the derivative of the following functions.

f(t) = t⁵/³e^t

Find and simplify the derivative of the following functions.

y = (3t−1)(2t−2)^-1

Find and simplify the derivative of the following functions.

h(x) = (x − 1)(x³+ x² + x+1)

Find and simplify the derivative of the following functions.

f(x) = e^x(x³ − 3x² + 6x − 6)