# 5.2: Sum and Difference Differentiation Rules

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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)=3x^{2}+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