Calculus Examples

$r = \log_{2} (- 5 s^{4})$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d s} (r) = \frac{d}{d s} (\log_{2} (- 5 s^{4}))$

Step 2

The derivative of $r$ with respect to $s$ is $r'$ .

$r'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

Differentiate using the chain rule, which states that $\frac{d}{d s} [f (g (s))]$ is $f' (g (s)) g' (s)$ where $f (s) = \log_{2} (s)$ and $g (s) = - 5 s^{4}$ .

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Step 3.1.1

To apply the Chain Rule, set $u$ as $- 5 s^{4}$ .

$\frac{d}{d u} [\log_{2} (u)] \frac{d}{d s} [- 5 s^{4}]$

Step 3.1.2

The derivative of $\log_{2} (u)$ with respect to $u$ is $\frac{1}{u \ln (2)}$ .

$\frac{1}{u \ln (2)} \frac{d}{d s} [- 5 s^{4}]$

Step 3.1.3

Replace all occurrences of $u$ with $- 5 s^{4}$ .

$\frac{1}{- 5 s^{4} \ln (2)} \frac{d}{d s} [- 5 s^{4}]$

Step 3.2

Differentiate.

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Step 3.2.1

Move the negative in front of the fraction.

$- \frac{1}{5 s^{4} \ln (2)} \frac{d}{d s} [- 5 s^{4}]$

Step 3.2.2

Since $- 5$ is constant with respect to $s$ , the derivative of $- 5 s^{4}$ with respect to $s$ is $- 5 \frac{d}{d s} [s^{4}]$ .

$- \frac{1}{5 s^{4} \ln (2)} (- 5 \frac{d}{d s} [s^{4}])$

Step 3.2.3

Simplify terms.

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Step 3.2.3.1

Multiply $- 5$ by $- 1$ .

$5 \frac{1}{5 s^{4} \ln (2)} \frac{d}{d s} [s^{4}]$

Step 3.2.3.2

Combine $5$ and $\frac{1}{5 s^{4} \ln (2)}$ .

$\frac{5}{5 s^{4} \ln (2)} \frac{d}{d s} [s^{4}]$

Step 3.2.3.3

Cancel the common factor of $5$ .

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Step 3.2.3.3.1

Cancel the common factor.

$\frac{5}{5 s^{4} \ln (2)} \frac{d}{d s} [s^{4}]$

Step 3.2.3.3.2

Rewrite the expression.

$\frac{1}{s^{4} \ln (2)} \frac{d}{d s} [s^{4}]$

Step 3.2.4

Differentiate using the Power Rule which states that $\frac{d}{d s} [s^{n}]$ is $n s^{n - 1}$ where $n = 4$ .

$\frac{1}{s^{4} \ln (2)} (4 s^{3})$

Step 3.2.5

Simplify terms.

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Step 3.2.5.1

Combine $4$ and $\frac{1}{s^{4} \ln (2)}$ .

$\frac{4}{s^{4} \ln (2)} s^{3}$

Step 3.2.5.2

Combine $\frac{4}{s^{4} \ln (2)}$ and $s^{3}$ .

$\frac{4 s^{3}}{s^{4} \ln (2)}$

Step 3.2.5.3

Cancel the common factor of $s^{3}$ and $s^{4}$ .

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Step 3.2.5.3.1

Factor $s^{3}$ out of $4 s^{3}$ .

$\frac{s^{3} \cdot 4}{s^{4} \ln (2)}$

Step 3.2.5.3.2

Cancel the common factors.

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Step 3.2.5.3.2.1

Factor $s^{3}$ out of $s^{4} \ln (2)$ .

$\frac{s^{3} \cdot 4}{s^{3} (s \ln (2))}$

Step 3.2.5.3.2.2

Cancel the common factor.

$\frac{s^{3} \cdot 4}{s^{3} (s \ln (2))}$

Step 3.2.5.3.2.3

Rewrite the expression.

$\frac{4}{s \ln (2)}$

Step 4

Reform the equation by setting the left side equal to the right side.

$r' = \frac{4}{s \ln (2)}$

Step 5

Replace $r'$ with $\frac{d r}{d s}$ .

$\frac{d r}{d s} = \frac{4}{s \ln (2)}$