Calculus Examples

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

Step 1

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

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

Step 2

Evaluate $\frac{d}{d x} [2 x \sin (x)]$ .

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

Multiply $\sin (x)$ by $1$ .

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

Step 3

Evaluate $\frac{d}{d x} [x^{2} \cos (x)]$ .

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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 $g (x) = \cos (x)$ .

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

Step 3.2

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

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

Step 3.3

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

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

Step 4

Simplify.

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Step 4.1

Apply the distributive property.

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

Step 4.2

Add $2 x \cos (x)$ and $\cos (x) (2 x)$ .

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Step 4.2.1

Move $\cos (x)$ .

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

Step 4.2.2

Add $2 x \cos (x)$ and $2 x \cos (x)$ .

$4 x \cos (x) + 2 \sin (x) + x^{2} (- \sin (x))$

Step 4.3

Reorder terms.

$4 x \cos (x) - x^{2} \sin (x) + 2 \sin (x)$