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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:

    \[\frac{d}{dx}[f(x)+g(x)]= \frac{d}{dx}[f(x)]+ \frac{d}{dx}ddx[g(x) \nonumber\]


    \[\frac{d}{dx}[f(x)−g(x)]=\frac{d}{dx}[f(x)]−\frac{d}{dx}[g(x)] \nonumber\]

    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:

    \[\frac{d}{dx}3x^2+2x]=\frac{d}{dx}[3x^2]+\frac{d}{dx}[2x] \nonumber\]

    = \[3 \frac{d}{dx}[x^2]+2 \frac{d}{dx}[x] \nonumber\]

    = \[3[2x]+2[1] \nonumber\]

    = \[6x+2 \nonumber\]


    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


    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 \[f^{\prime \prime}(5)=20\nonumber\]

    Review (Answers)

    To see the Review answers, open this PDF file and look for section 3.5.


    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

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