Calculus Examples

$x z e^{y z}$

Step 1

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

$x \frac{d}{d z} [z e^{y z}]$

Step 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 $f (z) = z$ and $g (z) = e^{y z}$ .

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

Step 3

Differentiate using the chain rule, which states that $\frac{d}{d z} [f (g (z))]$ is $f' (g (z)) g' (z)$ where $f (z) = e^{z}$ and $g (z) = y z$ .

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Step 3.1

To apply the Chain Rule, set $u$ as $y z$ .

$x (z (\frac{d}{d u} [e^{u}] \frac{d}{d z} [y z]) + e^{y z} \frac{d}{d z} [z])$

Step 3.2

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

$x (z (e^{u} \frac{d}{d z} [y z]) + e^{y z} \frac{d}{d z} [z])$

Step 3.3

Replace all occurrences of $u$ with $y z$ .

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

Step 4

Differentiate.

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Step 4.1

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

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

Step 4.2

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

$x (z (e^{y z} (y \cdot 1)) + e^{y z} \frac{d}{d z} [z])$

Step 4.3

Multiply $y$ by $1$ .

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

Step 4.4

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

$x (z (e^{y z} y) + e^{y z} \cdot 1)$

Step 4.5

Multiply $e^{y z}$ by $1$ .

$x (z (e^{y z} y) + e^{y z})$

Step 5

Simplify.

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Step 5.1

Apply the distributive property.

$x (z (e^{y z} y)) + x e^{y z}$

Step 5.2

Reorder terms.

$e^{y z} x y z + e^{y z} x$

Step 5.3

Reorder factors in $e^{y z} x y z + e^{y z} x$ .

$x y z e^{y z} + x e^{y z}$