Calculus Examples

$w = 2 z^{2} e^{z}$

Step 1

Differentiate both sides of the equation.

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

Step 2

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

$w'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

Since $2$ is constant with respect to $z$ , the derivative of $2 z^{2} e^{z}$ with respect to $z$ is $2 \frac{d}{d z} [z^{2} e^{z}]$ .

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

Step 3.2

Differentiate using the Product Rule which states that $\frac{d}{d z} [f (z) g (z)]$ is $f (z) \frac{d}{d z} [g (z)] + g (z) \frac{d}{d z} [f (z)]$ where $f (z) = z^{2}$ and $g (z) = e^{z}$ .

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

Step 3.3

Differentiate using the Exponential Rule which states that $\frac{d}{d z} [a^{z}]$ is $a^{z} \ln (a)$ where $a$ = $e$ .

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

Step 3.4

Differentiate using the Power Rule which states that $\frac{d}{d z} [z^{n}]$ is $n z^{n - 1}$ where $n = 2$ .

$2 (z^{2} e^{z} + e^{z} (2 z))$

Step 3.5

Simplify.

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Step 3.5.1

Apply the distributive property.

$2 (z^{2} e^{z}) + 2 (e^{z} (2 z))$

Step 3.5.2

Multiply $2$ by $2$ .

$2 z^{2} e^{z} + 4 e^{z} z$

Step 3.5.3

Reorder terms.

$2 e^{z} z^{2} + 4 e^{z} z$

Step 3.5.4

Reorder factors in $2 e^{z} z^{2} + 4 e^{z} z$ .

$2 z^{2} e^{z} + 4 z e^{z}$

Step 4

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

$w' = 2 z^{2} e^{z} + 4 z e^{z}$

Step 5

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

$\frac{d w}{d z} = 2 z^{2} e^{z} + 4 z e^{z}$