3. Techniques of Differentiation

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

Population growth Consider the following population functions.

c. Estimate the time when the instantaneous growth rate is greatest.

p(t) = 600 (t²+3/t²+9)

Power Rule for negative integers Use the Derivative Quotient Rule to prove the Power Rule for negative integers, that is,

d/dx (x⁻ᵐ) = −mx⁻ᵐ⁻¹

where m is a positive integer.

Use a graphing utility to graph the curve and the tangent line on the same set of axes.

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

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)

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

𝔂 = (x + 1)² (x² + 2x)

{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.

Use a graphing utility to graph the curve and the tangent line on the same set of axes.

y = 2x² / (3x - 1); a = 1

Derivatives Find and simplify the derivative of the following functions.

g(x) = e^x / x²-1

Find the derivative of the function.

$f\left(x\right)=\frac{2x-1}{x^3+2}$

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

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

Use differentiation to verify each equation.

d/dx(x / √1−x²) = 1 / (1−x²)^3/2.

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

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

Find the derivative the following ways:

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

g(t) = (t + 1)(t² - t + 1)

21–30. Derivatives

b. Evaluate f'(a) for the given values of a.

f(x) = 1/x+1; a = -1/2;5