Calculus Examples

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

Step 1

Differentiate.

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By the Sum Rule, the derivative of $x^{3} - x^{2} \cos (x) + 2 x \sin (x) + 2 \cos (x)$ with respect to $x$ is $\frac{d}{d x} [x^{3}] + \frac{d}{d x} [- x^{2} \cos (x)] + \frac{d}{d x} [2 x \sin (x)] + \frac{d}{d x} [2 \cos (x)]$ .

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

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

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

Step 2

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

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Since $- 1$ is constant with respect to $x$ , the derivative of $- x^{2} \cos (x)$ with respect to $x$ is $- \frac{d}{d x} [x^{2} \cos (x)]$ .

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

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

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

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

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

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

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

Step 3

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

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

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

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

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

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

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

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

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

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

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

Step 4

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

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

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

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

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

Multiply $- 1$ by $2$ .

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

Step 5

Simplify.

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Apply the distributive property.

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

Apply the distributive property.

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

Combine terms.

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Multiply $- 1$ by $- 1$ .

$3 x^{2} + 1 (x^{2} (\sin (x))) - (\cos (x) (2 x)) + 2 (x \cos (x)) + 2 \sin (x) - 2 \sin (x)$

Multiply $x^{2}$ by $1$ .

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

Multiply $2$ by $- 1$ .

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

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

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Move $\cos (x)$ .

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

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

$3 x^{2} + x^{2} \sin (x) + 0 + 2 \sin (x) - 2 \sin (x)$

Add $3 x^{2} + x^{2} \sin (x)$ and $0$ .

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

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

$3 x^{2} + x^{2} \sin (x) + 0$

Add $3 x^{2} + x^{2} \sin (x)$ and $0$ .

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