Review

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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

Average Value

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.

Previous Quizzes and Midterm

quiz1.pdf

quiz2.pdf

quiz3.pdf

quiz5.pdf

quiz6.pdf

quiz7.pdf

midterm.pdf

Questions

How do I ask a question?

What is the answer to 4b on the midterm?

What is the answer to 2a on Quiz 7?


Make the substitutions:

\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)