# 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)=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.