Calculus Examples

$y = x^{2} \sin (8 x)$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (x^{2} \sin (8 x))$

Step 2

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

$y'$

Step 3

Differentiate the right side of the equation.

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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 $f (x) = x^{2}$ and $g (x) = \sin (8 x)$ .

$x^{2} \frac{d}{d x} [\sin (8 x)] + \sin (8 x) \frac{d}{d x} [x^{2}]$

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 $f (x) = \sin (x)$ and $g (x) = 8 x$ .

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Step 3.2.1

To apply the Chain Rule, set $u$ as $8 x$ .

$x^{2} (\frac{d}{d u} [\sin (u)] \frac{d}{d x} [8 x]) + \sin (8 x) \frac{d}{d x} [x^{2}]$

Step 3.2.2

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

$x^{2} (\cos (u) \frac{d}{d x} [8 x]) + \sin (8 x) \frac{d}{d x} [x^{2}]$

Step 3.2.3

Replace all occurrences of $u$ with $8 x$ .

$x^{2} (\cos (8 x) \frac{d}{d x} [8 x]) + \sin (8 x) \frac{d}{d x} [x^{2}]$

Step 3.3

Differentiate.

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Step 3.3.1

Since $8$ is constant with respect to $x$ , the derivative of $8 x$ with respect to $x$ is $8 \frac{d}{d x} [x]$ .

$x^{2} (\cos (8 x) (8 \frac{d}{d x} [x])) + \sin (8 x) \frac{d}{d x} [x^{2}]$

Step 3.3.2

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

$x^{2} (\cos (8 x) (8 \cdot 1)) + \sin (8 x) \frac{d}{d x} [x^{2}]$

Step 3.3.3

Simplify the expression.

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Step 3.3.3.1

Multiply $8$ by $1$ .

$x^{2} (\cos (8 x) \cdot 8) + \sin (8 x) \frac{d}{d x} [x^{2}]$

Step 3.3.3.2

Move $8$ to the left of $\cos (8 x)$ .

$x^{2} (8 \cdot \cos (8 x)) + \sin (8 x) \frac{d}{d x} [x^{2}]$

Step 3.3.4

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

$x^{2} (8 \cos (8 x)) + \sin (8 x) (2 x)$

Step 3.3.5

Reorder terms.

$8 x^{2} \cos (8 x) + 2 x \sin (8 x)$

Step 4

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

$y' = 8 x^{2} \cos (8 x) + 2 x \sin (8 x)$

Step 5

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

$\frac{d y}{d x} = 8 x^{2} \cos (8 x) + 2 x \sin (8 x)$