Derivative Rules (College Board AP® Calculus BC): Study Guide

Power rule

How do I differentiate powers of x?

• Powers of $x$ are differentiated according to the following formula:

  • If$f\left(x\right)=x^n$ then$f\text{'}\left(x\right)=n·x^{n-1}$ 

  • This applies where $n\in \mathrm{ℝ}$ (real)

• This can also be written using the notation for differentiation with respect to $x$

  • $\frac{d}{dx}\left(x^n\right)=n·x^{n-1}$

• For example, if $f\left(x\right)=x^7$

  • $f\text{'}\left(x\right)=7x^{7-1}=7x^6$

• Be extra careful with fractional or negative powers, for example:

  • If $g\left(x\right)=x^{\frac{2}{3}}$ then $g\text{'}\left(x\right)=\frac{2}{3}{x}^{\frac{2}{3}-1}=\frac{2}{3}{x}^{-\frac{1}{3}}$

  • If $h\left(x\right)=x^{-3}$ then $h\text{'}\left(x\right)=-3x^{-3-1}=-3x^{-4}$

• This is much quicker than using the definition of a derivative, $\underset{h\to 0}{\lim }\frac{f\left(x+h\right)-f\left(x\right)}{h}$

  • However you should still be able to use this definition to find a derivative

Derivatives of sums, differences and constant multiples

How do I differentiate sums and differences?

• To differentiate a sum or difference of functions:

  • Differentiate each function individually

  • Take the sum or difference of the derivatives

• This can be written as:

  • $\frac{d}{dx}\left(f\left(x\right)\pm g\left(x\right)\right)=\frac{d}{dx}\left(f\left(x\right)\right)\pm \frac{d}{dx}\left(g\left(x\right)\right)$

• For example, if $f\left(x\right)=x^3+x^7-x^{12}+x^{\frac{1}{2}}$

  • Then $f^{\text{'}}\left(x\right)=3x^2+7x^6-12x^{11}+\frac{1}{2}{x}^{-\frac{1}{2}}$

How do I differentiate functions that have been multiplied by a constant?

• To differentiate a function that has been multiplied by a constant:

  • Differentiate the function

  • Multiply it by the constant

• This can be written as:

  • $\frac{d}{dx}\left(k·f\left(x\right)\right)=k·\frac{d}{dx}\left(f\left(x\right)\right)$

• Constant multiples of powers of $x$ are differentiated according to the following formula:

  • If$f\left(x\right)=a{x}^n$ then$f^{\text{'}}\left(x\right)=a·n{x}^{n-1}$ 

• For example, if $f\left(x\right)=12x^4$

  • $f^{\text{'}}\left(x\right)=12\times 4x^3=48x^3$

• Be careful with negative numbers

  • If $g\left(x\right)=-3x^{-7}$ then $g^{\text{'}}\left(x\right)=-3\times -7x^{-8}=21x^{-8}$

• This can then be applied to sums and differences

  • E.g. If $h\left(x\right)=2x^4+7x^{-5}-3x^8$

  • Then $h^{\text{'}}\left(x\right)=8x^3-35x^{-6}-24x^7$

  • This is especially useful when differentiating polynomials

How do I differentiate linear and constant functions?

• If $f\left(x\right)=ax$ then $f^{\text{'}}\left(x\right)=a$

  • You can derive this by writing the function as $f\left(x\right)=a{x}^1$

    • $f\text{'}\left(x\right)=1·a{x}^0=a$

  • You can also see this graphically by considering the slope of a straight line

    • It is a constant value

  • E.g. If $f\left(x\right)=4x$ then $f^{\text{'}}\left(x\right)=4$

    • This can also be seen graphically as the slope of the line $y=4x$ is $4$

• If $g\left(x\right)=a$ then $g\text{'}\left(x\right)=0$

  • You can derive this by taking the limit of the difference quotient formula

    • $g\text{'}\left(x\right)=\underset{h\to 0}{\lim }\frac{g\left(x+h\right)-g\left(x\right)}{h}=\underset{h\to 0}{\lim }\frac{a-a}{h}=0$

  • You can also see this graphically by considering the slope of a horizontal line

    • It is zero

  • E.g. If $g\left(x\right)=2$ then $g^{\text{'}}\left(x\right)=0$

Simplifying expressions to find derivatives

How do I simplify an expression before differentiating?

• If the function is not simply a sum of multiples of $x^n$, it may need to be simplified before it can be differentiated

• You may need to expand

  • E.g. $f\left(x\right)=\left(x+2\right)\left(2x^2+5\right)$

  • Expand the brackets

    • $f\left(x\right)=2x^3+4x^2+5x+10$

  • This is now a sum of multiples of $x^n$

  • Differentiate each term

    • $f^{\text{'}}\left(x\right)=6x^2+8x+5$

• You may need to rewrite the expression as a power of $x$ using laws of exponents

  • E.g. $g\left(x\right)=\frac{9x^4\times x^3}{x^2}$

  • Simplify using laws of exponents

    • $g\left(x\right)=\frac{9x^7}{x^2}=9x^5$

    • This can now be differentiated

      • $g^{\text{'}}\left(x\right)=45x^4$

  • Also consider $h\left(x\right)=\frac{2}{\sqrt{x}}$

  • Rewrite using laws of exponents

    • $h\left(x\right)=\frac{2}{x^{{\displaystyle \frac{1}{2}}}}=2x^{-\frac{1}{2}}$

  • This can now be differentiated

    • $h^{\text{'}}\left(x\right)=-x^{-\frac{3}{2}}$