Algebra Examples

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

Step 1

Differentiate both sides of the equation.

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

Step 2

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

$1$

Step 3

Differentiate the right side of the equation.

Tap for more steps...

Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

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

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

Step 3.5

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

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

Step 3.6

Rewrite $\frac{d}{d z} [c]$ as $c'$ .

$w^{\frac{3}{2}} (e^{w} c')$

Step 3.7

Reorder the factors of $w^{\frac{3}{2}} (e^{w} c')$ .

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

Step 4

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

$1 = e^{w} w^{\frac{3}{2}} c'$

Step 5

Solve for $c'$ .

Tap for more steps...

Step 5.1

Rewrite the equation as $e^{w} w^{\frac{3}{2}} c' = 1$ .

$e^{w} w^{\frac{3}{2}} c' = 1$

Step 5.2

Divide each term in $e^{w} w^{\frac{3}{2}} c' = 1$ by $e^{w} w^{\frac{3}{2}}$ and simplify.

Tap for more steps...

Step 5.2.1

Divide each term in $e^{w} w^{\frac{3}{2}} c' = 1$ by $e^{w} w^{\frac{3}{2}}$ .

$\frac{e^{w} w^{\frac{3}{2}} c'}{e^{w} w^{\frac{3}{2}}} = \frac{1}{e^{w} w^{\frac{3}{2}}}$

Step 5.2.2

Simplify the left side.

Tap for more steps...

Step 5.2.2.1

Cancel the common factor.

$\frac{e^{w} w^{\frac{3}{2}} c'}{e^{w} w^{\frac{3}{2}}} = \frac{1}{e^{w} w^{\frac{3}{2}}}$

Step 5.2.2.2

Rewrite the expression.

$\frac{w^{\frac{3}{2}} c'}{w^{\frac{3}{2}}} = \frac{1}{e^{w} w^{\frac{3}{2}}}$

Step 5.2.2.3

Cancel the common factor.

$\frac{w^{\frac{3}{2}} c'}{w^{\frac{3}{2}}} = \frac{1}{e^{w} w^{\frac{3}{2}}}$

Step 5.2.2.4

Divide $c'$ by $1$ .

$c' = \frac{1}{e^{w} w^{\frac{3}{2}}}$

Step 6

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

$\frac{d c}{d z} = \frac{1}{e^{w} w^{\frac{3}{2}}}$