Calculus Examples

${(x^{2} y^{3} + 3 x z + y^{2} z^{3} + 11)}^{3}$

Step 1

Differentiate using the chain rule, which states that $\frac{d}{d z} [f (g (z))]$ is $f' (g (z)) g' (z)$ where $f (z) = z^{3}$ and $g (z) = x^{2} y^{3} + 3 x z + y^{2} z^{3} + 11$ .

Tap for more steps...

Step 1.1

To apply the Chain Rule, set $u$ as $x^{2} y^{3} + 3 x z + y^{2} z^{3} + 11$ .

$\frac{d}{d u} [u^{3}] \frac{d}{d z} [x^{2} y^{3} + 3 x z + y^{2} z^{3} + 11]$

Step 1.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 3$ .

$3 u^{2} \frac{d}{d z} [x^{2} y^{3} + 3 x z + y^{2} z^{3} + 11]$

Step 1.3

Replace all occurrences of $u$ with $x^{2} y^{3} + 3 x z + y^{2} z^{3} + 11$ .

$3 {(x^{2} y^{3} + 3 x z + y^{2} z^{3} + 11)}^{2} \frac{d}{d z} [x^{2} y^{3} + 3 x z + y^{2} z^{3} + 11]$

Step 2

Differentiate.

Tap for more steps...

Step 2.1

By the Sum Rule, the derivative of $x^{2} y^{3} + 3 x z + y^{2} z^{3} + 11$ with respect to $z$ is $\frac{d}{d z} [x^{2} y^{3}] + \frac{d}{d z} [3 x z] + \frac{d}{d z} [y^{2} z^{3}] + \frac{d}{d z} [11]$ .

$3 {(x^{2} y^{3} + 3 x z + y^{2} z^{3} + 11)}^{2} (\frac{d}{d z} [x^{2} y^{3}] + \frac{d}{d z} [3 x z] + \frac{d}{d z} [y^{2} z^{3}] + \frac{d}{d z} [11])$

Step 2.2

Since $x^{2} y^{3}$ is constant with respect to $z$ , the derivative of $x^{2} y^{3}$ with respect to $z$ is $0$ .

$3 {(x^{2} y^{3} + 3 x z + y^{2} z^{3} + 11)}^{2} (0 + \frac{d}{d z} [3 x z] + \frac{d}{d z} [y^{2} z^{3}] + \frac{d}{d z} [11])$

Step 3

Evaluate $\frac{d}{d z} [3 x z]$ .

Tap for more steps...

Step 3.1

Since $3 x$ is constant with respect to $z$ , the derivative of $3 x z$ with respect to $z$ is $3 x \frac{d}{d z} [z]$ .

$3 {(x^{2} y^{3} + 3 x z + y^{2} z^{3} + 11)}^{2} (0 + 3 x \frac{d}{d z} [z] + \frac{d}{d z} [y^{2} z^{3}] + \frac{d}{d z} [11])$

Step 3.2

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

$3 {(x^{2} y^{3} + 3 x z + y^{2} z^{3} + 11)}^{2} (0 + 3 x \cdot 1 + \frac{d}{d z} [y^{2} z^{3}] + \frac{d}{d z} [11])$

Step 3.3

Multiply $3$ by $1$ .

$3 {(x^{2} y^{3} + 3 x z + y^{2} z^{3} + 11)}^{2} (0 + 3 x + \frac{d}{d z} [y^{2} z^{3}] + \frac{d}{d z} [11])$

Step 4

Evaluate $\frac{d}{d z} [y^{2} z^{3}]$ .

Tap for more steps...

Step 4.1

Since $y^{2}$ is constant with respect to $z$ , the derivative of $y^{2} z^{3}$ with respect to $z$ is $y^{2} \frac{d}{d z} [z^{3}]$ .

$3 {(x^{2} y^{3} + 3 x z + y^{2} z^{3} + 11)}^{2} (0 + 3 x + y^{2} \frac{d}{d z} [z^{3}] + \frac{d}{d z} [11])$

Step 4.2

Differentiate using the Power Rule which states that $\frac{d}{d z} [z^{n}]$ is $n z^{n - 1}$ where $n = 3$ .

$3 {(x^{2} y^{3} + 3 x z + y^{2} z^{3} + 11)}^{2} (0 + 3 x + y^{2} (3 z^{2}) + \frac{d}{d z} [11])$

Step 4.3

Move $3$ to the left of $y^{2}$ .

$3 {(x^{2} y^{3} + 3 x z + y^{2} z^{3} + 11)}^{2} (0 + 3 x + 3 y^{2} z^{2} + \frac{d}{d z} [11])$

Step 5

Since $11$ is constant with respect to $z$ , the derivative of $11$ with respect to $z$ is $0$ .

$3 {(x^{2} y^{3} + 3 x z + y^{2} z^{3} + 11)}^{2} (0 + 3 x + 3 y^{2} z^{2} + 0)$

Step 6

Simplify.

Tap for more steps...

Step 6.1

Combine terms.

Tap for more steps...

Step 6.1.1

Add $0$ and $3 x$ .

$3 {(x^{2} y^{3} + 3 x z + y^{2} z^{3} + 11)}^{2} (3 x + 3 y^{2} z^{2} + 0)$

Step 6.1.2

Add $3 x + 3 y^{2} z^{2}$ and $0$ .

$3 {(x^{2} y^{3} + 3 x z + y^{2} z^{3} + 11)}^{2} (3 x + 3 y^{2} z^{2})$

Step 6.2

Reorder terms.

$3 {(x^{2} y^{3} + 3 x z + y^{2} z^{3} + 11)}^{2} (3 y^{2} z^{2} + 3 x)$