2.2 Basic Differentiation Rules

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The following rules allow us the find the derivative of multiples, sums and differences of functions whose derivatives are already known.

Constant Multiple Rule

Constant Multiple Rule If $f(x)$ is differentiable and $c$ is any constant, then $[cf(x)]' = c f'(x)$

In words, the derivative of a constant times a function is the constant times the derivative of the function.

example 1 If $f(x) = 5x^2$ then we use the constant multiple rule with $c = 5$ and we get $f'(x) = 5(2x) = 10x.$

(problem 1a) Compute $\frac {d}{dx} \left (4x^2\right )$

The derivative of $x^2$ is $2x$

The derivative of $4x^2$ with respect to $x$ is $\answer {8x}$

(problem 1b) Compute $\frac {d}{dx} \left (5x^3\right )$

Use the Power Rule on $x^3$

The Power Rule says: $\frac {d}{dx} x^n = nx^{n-1}$

Don’t forget to multiply by $5$

The derivative of $5x^3$ with respect to $x$ is $\answer {15x^2}$

example 2 If $f(x) = \frac {x^3}{9}$ then we use the constant multiple rule with $c = \frac 19$ and we get $f'(x) = \tfrac 19 \cdot 3x^2 = \frac {x^2}{3}.$

(problem 2a) Compute $\frac {d}{dx} \left (\frac {x^2}{2}\right )$

The derivative of $x^2$ is $2x$

The derivative of $\frac {x^2}{2}$ with respect to $x$ is $\answer {x}$

(problem 2b) Compute $\frac {d}{dx} \left (8x^{1/2}\right )$

Use the Power Rule on $x^{1/2}$

The Power Rule says: $\frac {d}{dx} x^n = nx^{n-1}$

Don’t forget to multiply by $8$

The derivative of $8x^{1/2}$ with respect to $x$ is $\answer {4x^{-1/2}}$

(problem 2c) Compute $\frac {d}{dx} \left (\frac {x^5}{10}\right )$

Use the Power Rule on $x^5$

The Power Rule says: $\frac {d}{dx} x^n = nx^{n-1}$

Don’t forget to multiply by $1/10$

The derivative of $\frac {x^5}{10}$ with respect to $x$ is $\answer {\frac {x^4}{2}}$

example 3 If $f(x) = \frac {4}{x^2}$ then we rewrite $f(x)$ as $4x^{-2}$ and we use the constant multiple rule with $c = 4$ , giving $f'(x) = 4 (-2x^{-3}) = -8x^{-3} = -\frac {8}{x^3}.$

(problem 3a) Compute $\frac {d}{dx} \left (\frac {10}{x^5}\right )$

Rewrite using $\frac {1}{x^n} = x^{-n}$

Use the Power Rule on $x^{-5}$

The Power Rule says: $\frac {d}{dx} x^n = nx^{n-1}$

Don’t forget to multiply by 10

The derivative of $\frac {10}{x^5}$ with respect to $x$ is $\answer {-50x^{-6}}$

(problem 3b) Compute $\frac {d}{dx} \left (6\sqrt x\right )$

Rewrite using $\sqrt [n] x = x^{1/n}$

Use the Power Rule on $x^{1/2}$

The Power Rule says: $\frac {d}{dx} x^n = nx^{n-1}$

Don’t forget to multiply by 6

The derivative of $6\sqrt x$ with respect to $x$ is $\answer {3x^{-1/2}}$

Sum Rule

Sum Rule If $f(x)$ and $g(x)$ are differentiable functions then $[f(x) + g(x)]' = f'(x) + g'(x)$

In words, the derivative of a sum is the sum of the derivatives.

example 4 If $f(x) = x^3 + x^2$ then $f'(x) = 3x^2 + 2x$ .

(problem 4a) Compute $\frac {d}{dx} (x^5 + x^4)$

Use the Power Rule on each term

The Power Rule says: $\frac {d}{dx} x^n = nx^{n-1}$

The derivative of $x^5 + x^4$ with respect to $x$ is $\answer {5x^4 + 4x^3}$

(problem 4b) Compute $\frac {d}{dx} (x^{1000} + x^{100} + x^{10})$

The Sum Rule can be used on more than two terms

Use the Power Rule on each term

The Power Rule says: $\frac {d}{dx} x^n = nx^{n-1}$

The derivative of $x^{1000} + x^{100} + x^{10}$ with respect to $x$ is $\answer {1000x^{999} + 100x^{99} + 10x^9}$

example 5 If $f(x) = \frac {1}{x} + \frac {1}{x^2}$ then we rewrite $f(x)$ as $x^{-1} + x^{-2}$ and $f'(x) = -x^{-2} -2x^{-3} = -\frac {1}{x^2} - \frac {2}{x^3} = - \frac {x + 2}{x^3}.$

(problem 5a) Compute $\frac {d}{dx} \left (\frac {1}{x^3} + \frac {1}{x^4}\right )$

$\frac {1}{x^n} = x^{-n}$

Use the Power Rule on each term

The Power Rule says: $\frac {d}{dx} x^n = nx^{n-1}$

The derivative of $\frac {1}{x^3} + \frac {1}{x^4}$ with respect to $x$ is $\answer {-3x^{-4} -4x^{-5}}$

(problem 5b) Compute $\frac {d}{dx} \left (\frac {1}{x^5} + \frac {1}{x^6} + \frac {1}{x^7}\right )$

$\frac {1}{x^n} = x^{-n}$

The Sum Rule can be used on more than two terms

Use the Power Rule on each term

The Power Rule says: $\frac {d}{dx} x^n = nx^{n-1}$

The derivative of $\frac {1}{x^5} + \frac {1}{x^6} + \frac {1}{x^7}$ with respect to $x$ is $\answer {-5x^{-6} -6x^{-7} - 7x^{-8}}$

example 6 If $f(x) = \sqrt x + \sqrt [3] x$ then we rewrite $f(x)$ as $x^{1/2} + x^{1/3}$ and $\begin{align*} f'(x) &= \tfrac12 x^{-1/2} + \tfrac13 x^{-2/3} \\ &= \tfrac12 \cdot \frac{1}{x^{1/2}} + \tfrac13 \cdot \frac{1}{x^{2/3}}\\ &= \frac{1}{2\sqrt x} + \frac{1}{3\sqrt[3] {x^2}}. \end{align*}$

(problem 6a) Compute $\frac {d}{dx} \left (\sqrt [4] x + \sqrt [5] x\right )$

$\sqrt [n] x = x^{1/n}$

Use the Power Rule on each term

The Power Rule says: $\frac {d}{dx} x^n = nx^{n-1}$

The derivative of $\sqrt [4] x + \sqrt [5] x$ with respect to $x$ is $\answer {\frac 14 x^{-3/4} + \frac 15 x^{-4/5}}$

(problem 6b) Compute $\frac {d}{dx} \left (\sqrt [6] x + \sqrt [7] x + \sqrt [8] x\right )$

$\sqrt [n] x = x^{1/n}$

The Sum Rule can be used on more than two terms

Use the Power Rule on each term

The Power Rule says: $\frac {d}{dx} x^n = nx^{n-1}$

The derivative of $\sqrt [6] x + \sqrt [7] x + \sqrt [8] x$ with respect to $x$ is $\answer {\frac 16 x^{-5/6} + \frac 17 x^{-6/7} + \frac 18 x^{-7/8}}$

Difference Rule

Difference Rule If $f(x)$ and $g(x)$ are differentiable functions, then $[f(x) - g(x)]' = f'(x) - g'(x)$

In words, the derivative of a difference is the difference of the derivatives.

example 7 If $f(x) = x^5 - x$ then $f'(x) = 5x^4 - 1$ .

(problem 7a) Compute $\frac {d}{dx} \left (x^7 - x^4\right )$

Use the Power Rule on each term

The Power Rule says: $\frac {d}{dx} x^n = nx^{n-1}$

The derivative of $x^7 - x^4$ with respect to $x$ is $\answer {7x^6 - 4x^3}$

(problem 7b) Compute $\frac {d}{dx} \left (x^{1000} - x^{100} - x^{10}\right )$

The Difference Rule can be used on more than two terms

Use the Power Rule on each term

The Power Rule says: $\frac {d}{dx} x^n = nx^{n-1}$

The derivative of $x^{1000} - x^{100} - x^{10}$ with respect to $x$ is $\answer {1000x^{999} - 100x^{99} - 10x^9}$

(problem 7c) Compute $\frac {d}{dx} \left (4x^4 - 5x^3 + 8x^2 - 15x - 4\right )$

The Difference Rule can be used on more than two terms

The Sum, Difference and Constant Multiple Rules are often used together

Use the Power Rule on each term

The Power Rule says: $\frac {d}{dx} x^n = nx^{n-1}$

The derivative of $4x^4 - 5x^3 + 8x^2 - 15x - 4$ with respect to $x$ is $\answer {16x^3 - 15x^2 + 16x - 15}$

example 8 If $f(x) = \frac {1}{x^3} - \frac {1}{x^4}$ then we rewrite $f(x)$ as $x^{-3} - x^{-4}$ and $f'(x) = -3x^{-4} + 4x^{-5} = -\frac {3}{x^4} + \frac {4}{x^5} = \frac {4}{x^5}-\frac {3}{x^4} = \frac {4x^4 - 3x^5}{x^9} = \frac {4 - 3x}{x^5}$ .

(problem 8a) Compute $\frac {d}{dx} \left (\frac {1}{x^5} - \frac {1}{x}\right )$

$\frac {1}{x^n} = x^{-n}$

Use the Power Rule on each term

The Power Rule says: $\frac {d}{dx} x^n = nx^{n-1}$

The derivative of $\frac {1}{x^5} - \frac {1}{x}$ with respect to $x$ is $\answer {-5x^{-6} + x^{-2}}$

(problem 8b) Compute $\frac {d}{dx} \left (\frac {1}{x^2} - \frac {1}{x^6} - \frac {1}{x^7}\right )$

$\frac {1}{x^n} = x^{-n}$

The Difference Rule can be used on more than two terms

Use the Power Rule on each term

The Power Rule says: $\frac {d}{dx} x^n = nx^{n-1}$

The derivative of $\frac {1}{x^2} - \frac {1}{x^6} - \frac {1}{x^7}$ with respect to $x$ is $\answer {-2x^{-3} + 6x^{-7} + 7x^{-8}}$

example 9 If $f(x) = \sqrt x - \sqrt [4] {x^3}$ then we rewrite $f(x)$ as $x^{1/2} + x^{3/4}$ and $f'(x) = \tfrac 12 x^{-1/2} - \tfrac 34 x^{-1/4} = \tfrac 12 \cdot \frac {1}{x^{1/2}} - \tfrac 34 \cdot \frac {1}{x^{1/4}} = \frac {1}{2\sqrt x} - \frac {3}{4\sqrt [4] x}$ .

(problem 9a) Compute $\frac {d}{dx} \left (\sqrt [3] x - \sqrt [5] x\right )$

$\sqrt [n] x = x^{1/n}$

Use the Power Rule on each term

The Power Rule says: $\frac {d}{dx} x^n = nx^{n-1}$

The derivative of $\sqrt [3] x - \sqrt [5] x$ with respect to $x$ is $\answer {\frac 13 x^{-2/3} - \frac 15 x^{-4/5}}$

(problem 9b) Compute $\frac {d}{dx} (\sqrt [6] {x^5} - \sqrt [7] {x^6} + \sqrt [8] {x^7})$

$\sqrt [n] x^m = x^{m/n}$

The Sum and Difference Rules can be used on more than two terms

Use the Power Rule on each term

The Power Rule says: $\frac {d}{dx} x^n = nx^{n-1}$

The derivative of $\sqrt [6] {x^5} - \sqrt [7] {x^6} + \sqrt [8] {x^7}$ with respect to $x$ is $\answer {\frac 56 x^{-1/6} - \frac 67 x^{-1/7} + \frac 78 x^{-1/8}}$