Calculus Examples

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

Step 1

Rewrite ${(x - 1)}^{2}$ as $(x - 1) (x - 1)$ .

$\frac{d}{d x} [(x - 1) (x - 1) \sin (x)]$

Step 2

Expand $(x - 1) (x - 1)$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

$\frac{d}{d x} [(x (x - 1) - 1 (x - 1)) \sin (x)]$

Step 2.2

Apply the distributive property.

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

Step 2.3

Apply the distributive property.

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

Step 3

Simplify and combine like terms.

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Step 3.1

Simplify each term.

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Step 3.1.1

Multiply $x$ by $x$ .

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

Step 3.1.2

Move $- 1$ to the left of $x$ .

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

Step 3.1.3

Rewrite $- 1 x$ as $- x$ .

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

Step 3.1.4

Rewrite $- 1 x$ as $- x$ .

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

Step 3.1.5

Multiply $- 1$ by $- 1$ .

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

Step 3.2

Subtract $x$ from $- x$ .

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

Step 4

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 $f (x) = x^{2} - 2 x + 1$ and $g (x) = \sin (x)$ .

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

Step 5

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

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

Step 6

Differentiate.

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Step 6.1

By the Sum Rule, the derivative of $x^{2} - 2 x + 1$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 2 x] + \frac{d}{d x} [1]$ .

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

Step 6.2

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

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

Step 6.3

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

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

Step 6.4

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

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

Step 6.5

Multiply $- 2$ by $1$ .

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

Step 6.6

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$(x^{2} - 2 x + 1) \cos (x) + \sin (x) (2 x - 2 + 0)$

Step 6.7

Add $2 x - 2$ and $0$ .

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

Step 7

Simplify.

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Step 7.1

Apply the distributive property.

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

Step 7.2

Apply the distributive property.

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

Step 7.3

Combine terms.

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Step 7.3.1

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

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

Step 7.3.2

Move $- 2$ to the left of $\sin (x)$ .

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

Step 7.4

Reorder terms.

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