2.3: Differentiation Rules
- Page ID
- 204095
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)2.3 Differentiation Rules
- Apply the constant, power, sum and difference, product, and quotient rules to calculate the derivatives of a variety of functions, including polynomials, rational functions, and products of functions.
- Use the extended power rule to find the derivatives of functions with negative exponents, demonstrating the correct application to both positive and negative integer exponents.
- Combine the constant, power, sum and difference, product, and quotient rules to calculate the derivatives of polynomial and rational functions.
Differentiation Formulas:
1. Derivative of a constant function:
\[
\frac{d}{dx}(c) = 0
\]
2. Power rule (for any real number \( n \)):
\[
\frac{d}{dx}(x^n) = nx^{n - 1}
\]
3. Constant multiple rule:
\[
\frac{d}{dx}[c f(x)] = c \frac{d}{dx} f(x)
\]
4. Sum rule:
\[
\frac{d}{dx}[f(x) + g(x)] = \frac{d}{dx} f(x) + \frac{d}{dx} g(x)
\]
5. Difference rule:
\[
\frac{d}{dx}[f(x) - g(x)] = \frac{d}{dx} f(x) - \frac{d}{dx} g(x)
\]
6. Product rule:
\[
\frac{d}{dx}[f(x)g(x)] = \frac{d}{dx}[f(x)] \cdot g(x) + f(x) \cdot \frac{d}{dx}[g(x)]
\]
7. Quotient rule:
\[
\frac{d}{dx} \left[ \frac{f(x)}{g(x)} \right]
= \frac{ \frac{d}{dx}[f(x)] \cdot g(x) - f(x) \cdot \frac{d}{dx}[g(x)] }{ [g(x)]^2 }
\]
Find the derivative of each function below.
- \( y = 7 \)
- \( y = x^5 \)
- \( y = 4x^9 - x^2 + 1 \)
- \( y = x^\pi \)
- \( y = x^2 - \dfrac{3}{x} \)
- \( y = x^5(3x^2 + 2x + 1) \)
- \( y = x^4 \left( \dfrac{10}{x^4} + \dfrac{1}{x^2} \right) \)
- \( y = \dfrac{8x^6}{3x^2 - 1} \)
- \( y = \dfrac{\sqrt{x} + 1}{x^3 - 4x^2 + 5} \)
Find the equation of the tangent line \( T(x) \) to the graph of the given function at the specified point.
(1) \( f(x) = x^2 - \dfrac{3}{x} \) at \( (3, 8) \)
(2) \( f(x) = x^4 \left( \dfrac{10}{x^4} + \dfrac{1}{x^2} \right) \) at \( (2, 14) \)
(3) \( f(x) = \dfrac{\sqrt{x} + 1}{x^3 - 4x^2 + 5} \) at \( (1, 1) \)
Determine all points on the graph of
\[
f(x) = x^3 - 6x^2 +9x
\]
for which:
(a) the tangent line is horizontal
(b) the tangent line has a slope of \( -3 \)