5.2: Sum and Difference Differentiation Rules
- Page ID
- 1238
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Based on your knowledge of the limit definition of the derivative of a function, and the properties of limits discussed in a previous concept, can you make a prediction at this time how the derivative of a sum or difference of two functions should be determined?
Differentiation of Sums and Differences
Here are the differentiation rules for the sum and difference of two functions:
\[\frac{d}{dx}[f(x)+g(x)]= \frac{d}{dx}[f(x)]+ \frac{d}{dx}ddx[g(x) \nonumber\]
and
\[\frac{d}{dx}[f(x)−g(x)]=\frac{d}{dx}[f(x)]−\frac{d}{dx}[g(x)] \nonumber\]
In simpler notation \[(f±g)′=f′±g′ \nonumber\]
Using the limit properties of previous chapters should allow you to figure out why these differentiation rules apply.
You often need to apply multiple rules to find the derivative of a function. To find the derivative of f(x)=3x2+2x, you need to apply the sum of derivatives formula and the power rule:
\[\frac{d}{dx}3x^2+2x]=\frac{d}{dx}[3x^2]+\frac{d}{dx}[2x] \nonumber\]
= \[3 \frac{d}{dx}[x^2]+2 \frac{d}{dx}[x] \nonumber\]
= \[3[2x]+2[1] \nonumber\]
= \[6x+2 \nonumber\]
Examples
Example 1
Earlier, you were asked to make a prediction for the sum and differences of derivatves.
In a previous concept, you learned that if the limits exist:
\[ \displaystyle \lim_{x \to a} [f(x)±g(x)] = \lim_{x \to a} f(x)± \lim_{x \to a} g(x), \nonumber\]
Since the derivative of a function is defined by a limit, ddx[f(x)±g(x)] would be defined by limit applied to [f(x)±g(x)]. Work out the details to see that the above rules make sense.
Example 2
Given: t(x)=x−1, what is dt/dx when x=0
By the difference rule:
\[ (x−1)′=(x)′−(1)′=0 \nonumber\]
\[x′=1 \nonumber\]..... By the power rule
\[1′=0 \nonumber\]..... The derivative of a constant = 0
So when we evaluate this at x=0, we get 1, since \[1−0=1 \nonumber\]
Example 3
Find the derivative: \[ f(x)=x^3−5x^2 \nonumber\]
Use the difference and power rules to help:
\[ \frac{d}{dx}[x^3−5x^2]=\frac{d}{dx}[x^3]−5\frac{d}{dx}[x^2] \nonumber\]
\[ =3x^2−5[2x] \nonumber\]
\[ =3x^2−10x \nonumber\]
Example 4
Given \[ a(x)=−\pi x^{−0.54}+6x^4 \nonumber\] What is \[\frac{d}{dx}a(x)? \nonumber\]
We'll use the sum and power rules:
\[ \frac{d}{dx}(−\pi x^{−0.54}+6x^4)=\frac{d}{dx}(−\pi x^{−0.54}+\frac{d}{dx}(6x^4) \nonumber\]…By the sum rule
\[= -\pi \frac{d}{dx}(x^{−0.54})+6 \frac{d}{dx}(x^4) \nonumber\] …By the Constant - function Product rule
\[=0.54 \pi x^{−1.54}+24x^3 \nonumber\]…By the power rule
Review
For #1-7, find the derivative using the sum/difference rule
1. \[y=\frac{1}{2}\left(x^{3}-2 x^{2}+1\right)\nonumber\]
2. \[y=\sqrt{2} x^{3}-\frac{1}{\sqrt{2}} x^{2}+2 x+\sqrt{2}\nonumber\]
3. \[y=a^{2}-b^{2}+x^{2}-a-b+x\nonumber\] (where $a, b$ are constants)
4. \[y=x^{-3}+\frac{1}{x^{7}}\nonumber\]
5. \[y=\sqrt{x}+\frac{1}{\sqrt{x}}\nonumber\]
6. \[f(x)=(-3 x+4)^{2}\nonumber\]
7. \[f(x)=-0.93 x^{10}+\left(\pi^{3} x\right)^{\frac{-5}{12}}\nonumber\]
8. What is \[\frac{d}{d x}(2 x+1)^{2} ?\nonumber\]
9. Given: \[a(x)=(-5 x+3)^{2}\nonumber\] What is \[\frac{d y}{d x} ?\nonumber\]
10. \[v(x)=-3 x^{3}+5 x^{2}-2 x-3\nonumber\] What is \[v^{\prime}(0) ?\nonumber\]
11. \[f(x)=2 x^{2}+3 x+1 .\nonumber\] Find \[f^{\prime}(x)\nonumber\]
12. \[f(x)=\frac{1}{\sqrt{x}}-\frac{1}{x} .\nonumber\] Find \[f^{\prime}(1)\nonumber\]
13. \[y=(x+1)(x+2) \cdot\nonumber\] Evaluate \[\frac{d y}{d x}\nonumber\] at \[x=-\frac{1}{2}\nonumber\]
14. \[f(x)=2 a x^{3}+x^{2} ; f^{\prime}(-2)=0 \nonumber\] Find a
15. \[f(x)=a\left(x^{2}-5\right) ;\nonumber\] find a so that \[f^{\prime \prime}(5)=20\nonumber\]
Vocabulary
Term | Definition |
---|---|
derivative | The derivative of a function is the slope of the line tangent to the function at a given point on the graph. Notations for derivative include f′(x), dydx, y′, dfdx and \frac{df(x)}{dx}. |
differentiable | A differentiable function is a function that has a derivative that can be calculated. |
Additional Resources
Video: Product Rule
Practice: Sum and Difference Differentiation Rules