Calculus Examples

$s = 0.007184 w^{0.425} h^{0.725}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d w} (s) = \frac{d}{d w} (0.007184 w^{0.425} h^{0.725})$

Step 2

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

$s'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

Since $0.007184 h^{0.725}$ is constant with respect to $w$ , the derivative of $0.007184 w^{0.425} h^{0.725}$ with respect to $w$ is $0.007184 h^{0.725} \frac{d}{d w} [w^{0.425}]$ .

$0.007184 h^{0.725} \frac{d}{d w} [w^{0.425}]$

Step 3.2

Differentiate using the Power Rule which states that $\frac{d}{d w} [w^{n}]$ is $n w^{n - 1}$ where $n = 0.425$ .

$0.007184 h^{0.725} (0.425 w^{- 0.575})$

Step 3.3

Multiply $0.425$ by $0.007184$ .

$0.0030532 h^{0.725} w^{- 0.575}$

Step 3.4

Simplify.

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Step 3.4.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$0.0030532 h^{0.725} \frac{1}{w^{0.575}}$

Step 3.4.2

Combine terms.

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Step 3.4.2.1

Combine $0.0030532$ and $\frac{1}{w^{0.575}}$ .

$h^{0.725} \frac{0.0030532}{w^{0.575}}$

Step 3.4.2.2

Combine $h^{0.725}$ and $\frac{0.0030532}{w^{0.575}}$ .

$\frac{h^{0.725} \cdot 0.0030532}{w^{0.575}}$

Step 3.4.2.3

Move $0.0030532$ to the left of $h^{0.725}$ .

$\frac{0.0030532 h^{0.725}}{w^{0.575}}$

Step 4

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

$s' = \frac{0.0030532 h^{0.725}}{w^{0.575}}$

Step 5

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

$\frac{d s}{d w} = \frac{0.0030532 h^{0.725}}{w^{0.575}}$