Explain the difference between the average rate of change and the instantaneous rate of change of a function f.

Use the precise definition of a limit to prove the following limits. Specify a relationship between ε and δ that guarantees the limit exists.

lim x→a (mx+b)=ma+b, for any constants a, b, and m

72–76. Tangent lines Find an equation of the line tangent to each of the following curves at the given point.

y = 3x³+ sin x; (0, 0)

The function $s(t)$ represents the position of an object at time t moving along a line. Suppose $s(2)=136$ and $s(3)=156$ . Find the average velocity of the object over the interval of time $[2, 3]$ .

Evaluate and simplify y'.

y = 2x^√2

Evaluate and simplify y'.

y = 4u²+u / 8u+1

Evaluate and simplify y'.

y = (x²+1)³ / (x⁴+7)⁸(2x+1)⁷

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=

Higher-order derivatives Find and simplify y''.

y = 2^x x

9–61. Evaluate and simplify y'.

y = e^2θ

9–61. Evaluate and simplify y'.

y = e^sin x+2x+1

9–61. Evaluate and simplify y'.

y = e^tan x (tan x−1)

9–61. Evaluate and simplify y'.

y = x^√x+1

9–61. Evaluate and simplify y'.

y = e^sin (cosx)

9–61. Evaluate and simplify y'.

y = e^6x sin x

9–61. Evaluate and simplify y'.

y = 2^x²−x

9–61. Evaluate and simplify y'.

y = 10^sin x+sin¹⁰x

Use differentiation to verify each equation.

d/dx (x⁴ − ln(x⁴ + 1))=4x⁷ / (1 + x⁴).

Evaluate and simplify y'.

y = 4x⁴ ln x − x⁴

Evaluate and simplify y'.

y = ln w / w⁵

Find f′(1) when f(x) = x^(1/x).

Evaluate and simplify y'.

y = ln |sec 3x|

5-8. Use differentiation to verify each equation.

d/dx (tan³ x-3 tan x+3x) = 3 tan⁴x

9–61. Evaluate and simplify y'.

y = 5t² sin t

9–61. Evaluate and simplify y'.

y = (sin x / cos x+1)^1/3

9–61. Evaluate and simplify y'.

y = tan^−1 √t²−1

9–61. Evaluate and simplify y'.

y = csc⁵ 3x