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.

The 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.

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

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 $\answer [given]{8x}$

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 $\answer [given]{15x^2}$

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

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 $\answer [given]{x}$

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 $\answer [given]{4x^{-1/2}}$

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 $\answer [given]{\frac {x^4}{2}}$

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 $f'(x) = 4 (-2x^{-3}) = -8x^{-3} = -\frac {8}{x^3}.$

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 $\answer [given]{-50x^{-6}}$

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 $\answer [given]{3x^{-1/2}}$

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.

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

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 $\answer [given]{5x^4 + 4x^3}$

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 $\answer [given]{1000x^{999} + 100x^{99} + 10x^9}$

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

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 $\answer [given]{-3x^{-4} -4x^{-5}}$

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 $\answer [given]{-5x^{-6} -6x^{-7} - 7x^{-8}}$

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

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 $\answer [given]{\frac 14 x^{-3/4} + \frac 15 x^{-4/5}}$

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 $\answer [given]{\frac 16 x^{-5/6} + \frac 17 x^{-6/7} + \frac 18 x^{-7/8}}$

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.

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

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 $\answer [given]{7x^6 - 4x^3}$

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 $\answer [given]{1000x^{999} - 100x^{99} - 10x^9}$

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 $\answer [given]{16x^3 - 15x^2 + 16x - 15}$

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

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 $\answer [given]{-5x^{-6} + x^{-2}}$

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 $\answer [given]{-2x^{-3} + 6x^{-7} + 7x^{-8}}$

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

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 $\answer [given]{\frac 13 x^{-2/3} - \frac 15 x^{-4/5}}$

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 $\answer [given]{\frac 56 x^{-1/6} - \frac 67 x^{-1/7} + \frac 78 x^{-1/8}}$