# Review

## Change

• The change in f(x) from (x_1, y_1) to (x_2, y_2) is y_2 - y_1.

• The average rate of change in f(x) from (x_1, y_1) to (x_2, y_2) is \frac{y_2 - y_1}{x_2 - x_1}.

• The derivative of f(x) at x is f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}.

## Derivative Formulas

1. Constant Rule: f(x) = c \quad f'(x) = 0

2. Constant Multiplier Rule: f(x) = c \cdot g(x) \quad f'(x) = c \cdot g'(x)

3. Power Rule: f(x) = x^n \quad f'(x) = nx^{n-1}

4. Sum Rule: f(x) = g(x) + h(x) \quad f'(x) = g'(x) + h'(x)

5. Exponential Rule: f(x) = b^x \quad f'(x) = \ln(b) b^x

6. Logarithm Rule: f(x) = \ln(x) \quad f'(x) = \frac{1}{x}

7. Product Rule: f(x) = g(x) \cdot h(x) \quad f'(x) = g(x) \cdot h'(x) + g'(x) \cdot h(x)

8. Chain Rule: f(x) = g \circ h (x) \quad f'(x) = g' \circ h (x) h'(x)

## Anti-derivatives

• If F'(x) = f(x), then the antiderivative of f(x) is \int f(x) dx = F(x) + c. If a value of F(x_0) is known for a particular x_0, then we may solve for c and find the specific antiderivative.

• If F'(x) = f(x), then the definite integral from a to b of f(x) is \int_a^b f(x) dx = F(b) - F(a).

## Average Value

• The average value of f(x) from a to b is \frac{\int_a^b f(x) dx}{b - a}.

## Practice Exercises

Using the derivative formulas above, compute each of the following derivatives in questions 1-10. You may replace the letters serving as constants with your favorite numbers (a,b,c,d,e, and f) first if you wish. In particular, note that making some constants equal to one will greatly simplify the problem.

1. y(x) = ax^b

2. y(x) = \frac{ax^b + cx^d}{ex^f}

3. y(x) = e^{ax^b}

4. y(x) = \ln{ax^b}

5. y(x) = \frac{a}{b + ce^{dx}}

6. y(x) = ab^{cx}

7. y(x) = a(b + cx^d)^e

8. y(x) = \frac{ax^b + c}{dx^e + f}

9. y(x) = (a + b\ln(cx))(d + e\ln(fx))

10. y(x) = a

11. If the units of x are input and the units of f(x) are output, what are the units of f'(x)?

12. For some practice with word problems, try 1ab, 2-4 on page 224 of our textbook as well as the midterm review in our class packet.

# Questions

How do I ask a question?

• Press "edit" near the top of the page and ask your question at the bottom like this one.

What is the answer to 4b on the midterm?

• C(x) = \frac{57.6}{1 + 18.4e^{-0.3x}} + 9.7
C'(x) = \frac{57.6 \cdot 18.4 \cdot 0.3e^{-0.3x}}{(1 + 18.4e^{-0.3x})^2}

What is the answer to 2a on Quiz 7?

• \int x^2 \sqrt{x + 1} dx

Make the substitutions:

• u=x + 1

du = dx

x = u-1

x^2 = (u-1)^2

\int x^2 \sqrt{x + 1} dx = \int (u-1)^2 \sqrt{u}du = \int (u^2 -2u + 1)\sqrt{u}du = \int u^\frac{5}{2} du -2 \int u^\frac{3}{2} du + \int u^\frac{1}{2} du = \frac{2u^\frac{7}{2}}{7} + \frac{2u^\frac{5}{2}}{5} + \frac{2u^\frac{3}{2}}{3} + c = \frac{2(x+1)^\frac{7}{2}}{7} + \frac{2(x+1)^\frac{5}{2}}{5} + \frac{2(x+1)^\frac{3}{2}}{3} + c

NUWiki: Math131 (last edited 2011-08-07 18:22:31 by JasonRibeiro)