Calculus Examples

$y = {(x^{2} + 2)}^{2} {(x^{4} + 4)}^{4}$

Step 1

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = {(x^{2} + 2)}^{2}$ and $g (x) = {(x^{4} + 4)}^{4}$ .

${(x^{2} + 2)}^{2} \frac{d}{d x} [{(x^{4} + 4)}^{4}] + {(x^{4} + 4)}^{4} \frac{d}{d x} [{(x^{2} + 2)}^{2}]$

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{4}$ and $g (x) = x^{4} + 4$ .

Tap for more steps...

Step 2.1

To apply the Chain Rule, set $u_{1}$ as $x^{4} + 4$ .

${(x^{2} + 2)}^{2} (\frac{d}{d u_{1}} [{u_{1}}^{4}] \frac{d}{d x} [x^{4} + 4]) + {(x^{4} + 4)}^{4} \frac{d}{d x} [{(x^{2} + 2)}^{2}]$

Step 2.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{1}} [{u_{1}}^{n}]$ is $n {u_{1}}^{n - 1}$ where $n = 4$ .

${(x^{2} + 2)}^{2} (4 {u_{1}}^{3} \frac{d}{d x} [x^{4} + 4]) + {(x^{4} + 4)}^{4} \frac{d}{d x} [{(x^{2} + 2)}^{2}]$

Step 2.3

Replace all occurrences of $u_{1}$ with $x^{4} + 4$ .

${(x^{2} + 2)}^{2} (4 {(x^{4} + 4)}^{3} \frac{d}{d x} [x^{4} + 4]) + {(x^{4} + 4)}^{4} \frac{d}{d x} [{(x^{2} + 2)}^{2}]$

Step 3

Differentiate.

Tap for more steps...

Step 3.1

Move $4$ to the left of ${(x^{2} + 2)}^{2}$ .

$4 \cdot {(x^{2} + 2)}^{2} {(x^{4} + 4)}^{3} \frac{d}{d x} [x^{4} + 4] + {(x^{4} + 4)}^{4} \frac{d}{d x} [{(x^{2} + 2)}^{2}]$

Step 3.2

By the Sum Rule, the derivative of $x^{4} + 4$ with respect to $x$ is $\frac{d}{d x} [x^{4}] + \frac{d}{d x} [4]$ .

$4 {(x^{2} + 2)}^{2} {(x^{4} + 4)}^{3} (\frac{d}{d x} [x^{4}] + \frac{d}{d x} [4]) + {(x^{4} + 4)}^{4} \frac{d}{d x} [{(x^{2} + 2)}^{2}]$

Step 3.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

$4 {(x^{2} + 2)}^{2} {(x^{4} + 4)}^{3} (4 x^{3} + \frac{d}{d x} [4]) + {(x^{4} + 4)}^{4} \frac{d}{d x} [{(x^{2} + 2)}^{2}]$

Step 3.4

Since $4$ is constant with respect to $x$ , the derivative of $4$ with respect to $x$ is $0$ .

$4 {(x^{2} + 2)}^{2} {(x^{4} + 4)}^{3} (4 x^{3} + 0) + {(x^{4} + 4)}^{4} \frac{d}{d x} [{(x^{2} + 2)}^{2}]$

Step 3.5

Simplify the expression.

Tap for more steps...

Step 3.5.1

Add $4 x^{3}$ and $0$ .

$4 {(x^{2} + 2)}^{2} {(x^{4} + 4)}^{3} (4 x^{3}) + {(x^{4} + 4)}^{4} \frac{d}{d x} [{(x^{2} + 2)}^{2}]$

Step 3.5.2

Multiply $4$ by $4$ .

$16 {(x^{2} + 2)}^{2} {(x^{4} + 4)}^{3} x^{3} + {(x^{4} + 4)}^{4} \frac{d}{d x} [{(x^{2} + 2)}^{2}]$

Step 4

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

Tap for more steps...

Step 4.1

To apply the Chain Rule, set $u_{2}$ as $x^{2} + 2$ .

$16 {(x^{2} + 2)}^{2} {(x^{4} + 4)}^{3} x^{3} + {(x^{4} + 4)}^{4} (\frac{d}{d u_{2}} [{u_{2}}^{2}] \frac{d}{d x} [x^{2} + 2])$

Step 4.2

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

$16 {(x^{2} + 2)}^{2} {(x^{4} + 4)}^{3} x^{3} + {(x^{4} + 4)}^{4} (2 u_{2} \frac{d}{d x} [x^{2} + 2])$

Step 4.3

Replace all occurrences of $u_{2}$ with $x^{2} + 2$ .

$16 {(x^{2} + 2)}^{2} {(x^{4} + 4)}^{3} x^{3} + {(x^{4} + 4)}^{4} (2 (x^{2} + 2) \frac{d}{d x} [x^{2} + 2])$

Step 5

Differentiate.

Tap for more steps...

Step 5.1

Move $2$ to the left of ${(x^{4} + 4)}^{4}$ .

$16 {(x^{2} + 2)}^{2} {(x^{4} + 4)}^{3} x^{3} + 2 \cdot {(x^{4} + 4)}^{4} (x^{2} + 2) \frac{d}{d x} [x^{2} + 2]$

Step 5.2

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

$16 {(x^{2} + 2)}^{2} {(x^{4} + 4)}^{3} x^{3} + 2 {(x^{4} + 4)}^{4} (x^{2} + 2) (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [2])$

Step 5.3

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

$16 {(x^{2} + 2)}^{2} {(x^{4} + 4)}^{3} x^{3} + 2 {(x^{4} + 4)}^{4} (x^{2} + 2) (2 x + \frac{d}{d x} [2])$

Step 5.4

Since $2$ is constant with respect to $x$ , the derivative of $2$ with respect to $x$ is $0$ .

$16 {(x^{2} + 2)}^{2} {(x^{4} + 4)}^{3} x^{3} + 2 {(x^{4} + 4)}^{4} (x^{2} + 2) (2 x + 0)$

Step 5.5

Simplify the expression.

Tap for more steps...

Step 5.5.1

Add $2 x$ and $0$ .

$16 {(x^{2} + 2)}^{2} {(x^{4} + 4)}^{3} x^{3} + 2 {(x^{4} + 4)}^{4} (x^{2} + 2) (2 x)$

Step 5.5.2

Multiply $2$ by $2$ .

$16 {(x^{2} + 2)}^{2} {(x^{4} + 4)}^{3} x^{3} + 4 {(x^{4} + 4)}^{4} (x^{2} + 2) x$

Step 6

Simplify.

Tap for more steps...

Step 6.1

Factor $4 (x^{2} + 2) {(x^{4} + 4)}^{3} x$ out of $16 {(x^{2} + 2)}^{2} {(x^{4} + 4)}^{3} x^{3} + 4 {(x^{4} + 4)}^{4} (x^{2} + 2) x$ .

Tap for more steps...

Step 6.1.1

Factor $4 (x^{2} + 2) {(x^{4} + 4)}^{3} x$ out of $16 {(x^{2} + 2)}^{2} {(x^{4} + 4)}^{3} x^{3}$ .

$4 (x^{2} + 2) {(x^{4} + 4)}^{3} x (4 (x^{2} + 2) x^{2}) + 4 {(x^{4} + 4)}^{4} (x^{2} + 2) x$

Step 6.1.2

Factor $4 (x^{2} + 2) {(x^{4} + 4)}^{3} x$ out of $4 {(x^{4} + 4)}^{4} (x^{2} + 2) x$ .

$4 (x^{2} + 2) {(x^{4} + 4)}^{3} x (4 (x^{2} + 2) x^{2}) + 4 (x^{2} + 2) {(x^{4} + 4)}^{3} x (x^{4} + 4)$

Step 6.1.3

Factor $4 (x^{2} + 2) {(x^{4} + 4)}^{3} x$ out of $4 (x^{2} + 2) {(x^{4} + 4)}^{3} x (4 (x^{2} + 2) x^{2}) + 4 (x^{2} + 2) {(x^{4} + 4)}^{3} x (x^{4} + 4)$ .

$4 (x^{2} + 2) {(x^{4} + 4)}^{3} x (4 (x^{2} + 2) x^{2} + x^{4} + 4)$

Step 6.2

Reorder the factors of $4 (x^{2} + 2) {(x^{4} + 4)}^{3} x (4 (x^{2} + 2) x^{2} + x^{4} + 4)$ .

$4 x {(x^{4} + 4)}^{3} (4 (x^{2} + 2) x^{2} + x^{4} + 4) (x^{2} + 2)$