Calculus Examples

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

Step 1

Differentiate both sides of the equation.

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

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

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

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

Step 3.2

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

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Step 3.2.1

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

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

Step 3.2.2

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

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

Step 3.2.3

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

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

Step 3.3

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

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Step 3.3.1

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

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

Step 3.3.2

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

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

Step 3.3.3

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

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

Step 3.3.4

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

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

Step 3.3.5

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

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

Step 3.4

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

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Step 3.4.1

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

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

Step 3.4.2

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

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

Step 3.5

Simplify.

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Step 3.5.1

Apply the distributive property.

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

Step 3.5.2

Combine terms.

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Step 3.5.2.1

Multiply $- 1$ by $2$ .

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

Step 3.5.2.2

Subtract $2 x \sin (x)$ from $\sin (x) (2 x)$ .

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Step 3.5.2.2.1

Move $\sin (x)$ .

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

Step 3.5.2.2.2

Subtract $2 x \sin (x)$ from $2 x \sin (x)$ .

$x^{2} \cos (x) + 0 + 2 \cos (x) - 2 \cos (x)$

Step 3.5.2.3

Add $x^{2} \cos (x)$ and $0$ .

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

Step 3.5.2.4

Subtract $2 \cos (x)$ from $2 \cos (x)$ .

$x^{2} \cos (x) + 0$

Step 3.5.2.5

Add $x^{2} \cos (x)$ and $0$ .

$x^{2} \cos (x)$

Step 4

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

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

Step 5

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

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