# Review

## Definition of the Derivative

f'(x) = \lim_{h \rightarrow 0} \frac{f(x + h) - f(x)}{h}

The derivative at a point is the slope of the tangent line to the graph of the function at that point.

## Derivative Formulas

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

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

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

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

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

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

• f(x) = \log_{a} (x) \quad f'(x) = \frac{1}{x\ln(a)}

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

• Quotient Rule: f(x) = \frac{h(x)}{g(x)} \quad f'(x) = \frac{g(x) \cdot h'(x) - g'(x) \cdot h(x)}{(g(x))^2}

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

• Trig Rules:

• f(x) = \sin(x) \quad f'(x) = \cos(x)

f(x) = \cos(x) \quad f'(x) = -\sin(x)

f(x) = \tan(x) \quad f'(x) = \sec^2(x)

f(x) = \sec(x) \quad f'(x) = \sec(x)\tan(x)

f(x) = \csc(x) \quad f'(x) = -\csc(x)\cot(x)

f(x) = \cot(x) \quad f'(x) = -\csc^2(x)

• Inverse Trig Rules:

• f(x) = \sin^{-1}(x) \quad f'(x) = \frac{1}{\sqrt{1-x^2}}

f(x) = \cos^{-1}(x) \quad f'(x) = - \frac{1}{\sqrt{1-x^2}}

f(x) = \tan^{-1}(x) \quad f'(x) = \frac{1}{1+x^2}

f(x) = \sec^{-1}(x) \quad f'(x) = \frac{1}{|x|\sqrt{{x^2-1}}}

f(x) = \csc^{-1}(x) \quad f'(x) = - \frac{1}{|x|\sqrt{{x^2-1}}}

f(x) = \cot^{-1}(x) \quad f'(x) = - \frac{1}{1+x^2}

## 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)?

# Past Quizzes and Exams

 Quiz Solutions LaTeX source -

# 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.

NUWiki: Math1341 (last edited 2011-08-07 19:44:06 by JasonRibeiro)