Algebra Examples

$z = w^{\frac{3}{2}} (w + c e^{w})$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d c} (z) = \frac{d}{d c} (w^{\frac{3}{2}} (w + c e^{w}))$

Step 2

The derivative of $z$ with respect to $c$ is $z'$ .

$z'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

Since $w^{\frac{3}{2}}$ is constant with respect to $c$ , the derivative of $w^{\frac{3}{2}} (w + c e^{w})$ with respect to $c$ is $w^{\frac{3}{2}} \frac{d}{d c} [w + c e^{w}]$ .

$w^{\frac{3}{2}} \frac{d}{d c} [w + c e^{w}]$

Step 3.2

By the Sum Rule, the derivative of $w + c e^{w}$ with respect to $c$ is $\frac{d}{d c} [w] + \frac{d}{d c} [c e^{w}]$ .

$w^{\frac{3}{2}} (\frac{d}{d c} [w] + \frac{d}{d c} [c e^{w}])$

Step 3.3

Since $w$ is constant with respect to $c$ , the derivative of $w$ with respect to $c$ is $0$ .

$w^{\frac{3}{2}} (0 + \frac{d}{d c} [c e^{w}])$

Step 3.4

Add $0$ and $\frac{d}{d c} [c e^{w}]$ .

$w^{\frac{3}{2}} \frac{d}{d c} [c e^{w}]$

Step 3.5

Since $e^{w}$ is constant with respect to $c$ , the derivative of $c e^{w}$ with respect to $c$ is $e^{w} \frac{d}{d c} [c]$ .

$w^{\frac{3}{2}} (e^{w} \frac{d}{d c} [c])$

Step 3.6

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

$w^{\frac{3}{2}} (e^{w} \cdot 1)$

Step 3.7

Simplify the expression.

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Step 3.7.1

Multiply $e^{w}$ by $1$ .

$w^{\frac{3}{2}} e^{w}$

Step 3.7.2

Reorder the factors of $w^{\frac{3}{2}} e^{w}$ .

$e^{w} w^{\frac{3}{2}}$

Step 4

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

$z' = e^{w} w^{\frac{3}{2}}$

Step 5

Replace $z'$ with $\frac{d z}{d c}$ .

$\frac{d z}{d c} = e^{w} w^{\frac{3}{2}}$