Calculus Examples

$y = x^{9} \sin (x)$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (x^{9} \sin (x))$

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 $f (x) = x^{9}$ and $g (x) = \sin (x)$ .

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

Step 3.2

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

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

Step 3.3

Differentiate using the Power Rule.

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Step 3.3.1

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 9$ .

$x^{9} \cos (x) + \sin (x) (9 x^{8})$

Step 3.3.2

Reorder terms.

$x^{9} \cos (x) + 9 x^{8} \sin (x)$

Step 4

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

$y' = x^{9} \cos (x) + 9 x^{8} \sin (x)$

Step 5

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

$\frac{d y}{d x} = x^{9} \cos (x) + 9 x^{8} \sin (x)$