Calculus Examples

$y = {(x - 1)}^{2} + 1$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} ({(x - 1)}^{2} + 1)$

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

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

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

Step 3.2

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

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Step 3.2.1

Apply the distributive property.

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

Step 3.2.2

Apply the distributive property.

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

Step 3.2.3

Apply the distributive property.

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

Step 3.3

Simplify and combine like terms.

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Step 3.3.1

Simplify each term.

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Step 3.3.1.1

Multiply $x$ by $x$ .

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

Step 3.3.1.2

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

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

Step 3.3.1.3

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

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

Step 3.3.1.4

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

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

Step 3.3.1.5

Multiply $- 1$ by $- 1$ .

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

Step 3.3.2

Subtract $x$ from $- x$ .

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

Step 3.4

Differentiate.

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Step 3.4.1

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

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

Step 3.4.2

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

$2 x + \frac{d}{d x} [- 2 x] + \frac{d}{d x} [1] + \frac{d}{d x} [1]$

Step 3.5

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

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Step 3.5.1

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

$2 x - 2 \frac{d}{d x} [x] + \frac{d}{d x} [1] + \frac{d}{d x} [1]$

Step 3.5.2

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

$2 x - 2 \cdot 1 + \frac{d}{d x} [1] + \frac{d}{d x} [1]$

Step 3.5.3

Multiply $- 2$ by $1$ .

$2 x - 2 + \frac{d}{d x} [1] + \frac{d}{d x} [1]$

Step 3.6

Differentiate using the Constant Rule.

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Step 3.6.1

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

$2 x - 2 + 0 + \frac{d}{d x} [1]$

Step 3.6.2

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

$2 x - 2 + 0 + 0$

Step 3.7

Combine terms.

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Step 3.7.1

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

$2 x - 2 + 0$

Step 3.7.2

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

$2 x - 2$

Step 4

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

$y' = 2 x - 2$

Step 5

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

$\frac{d y}{d x} = 2 x - 2$