Calculus Examples

$y = x^{3} \sin^{3} (x^{3})$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (x^{3} \sin^{3} (x^{3}))$

Step 2

The derivative of $y$ with respect to $x$ is $y'$ .

$y'$

Step 3

Differentiate the right side of the equation.

Tap for more steps...

Step 3.1

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 $f (x) = x^{3}$ and $g (x) = \sin^{3} (x^{3})$ .

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

Step 3.2

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

Tap for more steps...

Step 3.2.1

To apply the Chain Rule, set $u_{1}$ as $\sin (x^{3})$ .

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

Step 3.2.2

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

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

Step 3.2.3

Replace all occurrences of $u_{1}$ with $\sin (x^{3})$ .

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

Step 3.3

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

Tap for more steps...

Step 3.3.1

To apply the Chain Rule, set $u_{2}$ as $x^{3}$ .

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

Step 3.3.2

The derivative of $\sin (u_{2})$ with respect to $u_{2}$ is $\cos (u_{2})$ .

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

Step 3.3.3

Replace all occurrences of $u_{2}$ with $x^{3}$ .

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

Step 3.4

Differentiate using the Power Rule.

Tap for more steps...

Step 3.4.1

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

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

Step 3.4.2

Multiply $3$ by $3$ .

$x^{3} (9 \sin^{2} (x^{3}) \cos (x^{3}) x^{2}) + \sin^{3} (x^{3}) \frac{d}{d x} [x^{3}]$

Step 3.5

Multiply $x^{3}$ by $x^{2}$ by adding the exponents.

Tap for more steps...

Step 3.5.1

Move $x^{2}$ .

$x^{2} x^{3} (9 \sin^{2} (x^{3}) \cos (x^{3})) + \sin^{3} (x^{3}) \frac{d}{d x} [x^{3}]$

Step 3.5.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x^{2 + 3} (9 \sin^{2} (x^{3}) \cos (x^{3})) + \sin^{3} (x^{3}) \frac{d}{d x} [x^{3}]$

Step 3.5.3

Add $2$ and $3$ .

$x^{5} (9 \sin^{2} (x^{3}) \cos (x^{3})) + \sin^{3} (x^{3}) \frac{d}{d x} [x^{3}]$

Step 3.6

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

$x^{5} (9 \sin^{2} (x^{3}) \cos (x^{3})) + \sin^{3} (x^{3}) (3 x^{2})$

Step 3.7

Reorder terms.

$9 x^{5} \sin^{2} (x^{3}) \cos (x^{3}) + 3 x^{2} \sin^{3} (x^{3})$

Step 4

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

$y' = 9 x^{5} \sin^{2} (x^{3}) \cos (x^{3}) + 3 x^{2} \sin^{3} (x^{3})$

Step 5

Replace $y'$ with $\frac{d y}{d x}$ .

$\frac{d y}{d x} = 9 x^{5} \sin^{2} (x^{3}) \cos (x^{3}) + 3 x^{2} \sin^{3} (x^{3})$