Constant Multiple Rule

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We find the derivative of a constant multiple of a function.

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

Compute $\frac {d}{dx} 4x$

The derivative of $x$ is $1$

The derivative of $4x$ with respect to $x$ is $\answer [given]{4}$

Compute $\frac {d}{dx} -7x$

The derivative of $x$ is $1$

The derivative of $-7x$ with respect to $x$ is $\answer [given]{-7}$

Compute $\frac {d}{dx} \frac {x}{2}$

The derivative of $x$ is $1$

The derivative of $\frac {x}{2}$ with respect to $x$ is $\answer [given]{1/2}$

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 $\frac {d}{dx} 5x^3$

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

Compute $\frac {d}{dx} 8x^{1/2}$

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 $\frac {d}{dx} \frac {x^5}{10}$

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 $\frac {d}{dx} \frac {10}{x^5}$

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 $\frac {d}{dx} 6\sqrt x$

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 $f(x) = -3\cos (x)$ then we use the constant multiple rule with $c = -3$ and we get $f'(x) = -3(-\sin (x)) = 3\sin (x).$

Compute $\frac {d}{dx} 5\sin (x)$

The derivative of $\sin (x)$ is $\cos (x)$

Don’t forget to multiply by 5

The derivative of $5\sin (x)$ with respect to $x$ is $\answer [given]{5\cos (x)}$

Compute $\frac {d}{dx} \pi \cos (x)$

The derivative of $\cos (x)$ is $-\sin (x)$

Don’t forget to multiply by $\pi$

The derivative of $\pi \cos (x)$ with respect to $x$ is $\answer [given]{-\pi \sin (x)}$

If $f(x) = \pi \tan ^{-1}(x)$ then we use the constant multiple rule with $c = \pi$ and we get $f'(x) = \pi \cdot \frac {1}{1+x^2}= \frac {\pi }{1+x^2}.$

Compute $\frac {d}{dx} 3\sin ^{-1}(x)$

The derivative of $\sin ^{-1}(x)$ is $\frac {1}{\sqrt {1-x^2}}$

Don’t forget to multiply by 3

The derivative of $3\sin ^{-1}(x)$ with respect to $x$ is $\answer [given]{\frac {3}{\sqrt {1-x^2}}}$

Compute $\frac {d}{dx} 2\cos ^{-1}(x)$

The derivative of $\cos ^{-1}(x)$ is $-\frac {1}{\sqrt {1-x^2}}$

Don’t forget to multiply by 2

The derivative of $2\cos ^{-1}(x)$ with respect to $x$ is $\answer [given]{-\frac {2}{\sqrt {1-x^2}}}$

Compute $\frac {d}{dx} 5\tan ^{-1}(x)$

The derivative of $\tan ^{-1}(x)$ is $\frac {1}{1+x^2}$

Don’t forget to multiply by 5

The derivative of $5\tan ^{-1}(x)$ with respect to $x$ is $\answer [given]{\frac {5}{1+x^2}}$

If $f(x) = \frac {2e^x}{5}$ then we use the constant multiple rule with $c = \frac 25$ and we get $f'(x) = \tfrac 25 e^x = \frac {2e^x}{5}.$

Compute $\frac {d}{dx} \frac {e^x}{2}$

The derivative of $e^x$ is $e^x$

Don’t forget to multiply by $1/2$

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

Compute $\frac {d}{dx} 3\cdot 2^x$

The derivative of $a^x$ is $a^x \ln (a)$

Don’t forget to multiply by $3$

The derivative of $3\cdot 2^x$ with respect to $x$ is $\answer [given]{3\cdot 2^x \ln (2)}$

If $f(x) = \frac {\ln (x)}{\ln (4)}$ then we use the constant multiple rule with $c = \frac {1}{\ln (4)}$ and we get $f'(x) = \tfrac {1}{\ln (4)} \cdot \frac {1}{x} = \frac {1}{x\ln (4)}.$

Compute $\frac {d}{dx} 8\ln (x)$

The derivative of $\ln (x)$ is $1/x$

Don’t forget to multiply by $8$

The derivative of $8\ln (x)$ with respect to $x$ is $\answer [given]{\frac {8}{x}}$

Compute $\frac {d}{dx} 2\log (x)$

The derivative of $\log _a(x)$ is $\frac {1}{x\ln (a)}$

$\log (x) = \log _10(x)$

Don’t forget to multiply by $2$

The derivative of $2\log (x)$ with respect to $x$ is $\answer [given]{\frac {2}{x\ln (10)}}$