Calculus Examples

$y = 3 x^{2} \sin (2 x)$

Step 1

Since $3$ is constant with respect to $x$ , the derivative of $3 x^{2} \sin (2 x)$ with respect to $x$ is $3 \frac{d}{d x} [x^{2} \sin (2 x)]$ .

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

Step 2

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 $f (x) = x^{2}$ and $g (x) = \sin (2 x)$ .

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

Step 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 $g (x) = 2 x$ .

Tap for more steps...

Step 3.1

To apply the Chain Rule, set $u$ as $2 x$ .

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

Step 3.2

The derivative of $\sin (u)$ with respect to $u$ is $\cos (u)$ .

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

Step 3.3

Replace all occurrences of $u$ with $2 x$ .

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

Step 4

Differentiate.

Tap for more steps...

Step 4.1

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

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

Step 4.2

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

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

Step 4.3

Simplify the expression.

Tap for more steps...

Step 4.3.1

Multiply $2$ by $1$ .

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

Step 4.3.2

Move $2$ to the left of $\cos (2 x)$ .

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

Step 4.4

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

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

Step 5

Simplify.

Tap for more steps...

Step 5.1

Apply the distributive property.

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

Step 5.2

Combine terms.

Tap for more steps...

Step 5.2.1

Multiply $2$ by $3$ .

$6 (x^{2} (\cos (2 x))) + 3 (\sin (2 x) (2 x))$

Step 5.2.2

Multiply $2$ by $3$ .

$6 x^{2} \cos (2 x) + 6 \sin (2 x) x$

Step 5.3

Reorder terms.

$6 x^{2} \cos (2 x) + 6 x \sin (2 x)$