Calculus Examples

$y = (x - 1) (x + 2)$

Step 1

Differentiate both sides of the equation.

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

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

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 $f (x) = x - 1$ and $g (x) = x + 2$ .

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

Step 3.2

Differentiate.

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Step 3.2.1

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

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

Step 3.2.2

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

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

Step 3.2.3

Since $2$ is constant with respect to $x$ , the derivative of $2$ with respect to $x$ is $0$ .

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

Step 3.2.4

Simplify the expression.

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Step 3.2.4.1

Add $1$ and $0$ .

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

Step 3.2.4.2

Multiply $x - 1$ by $1$ .

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

Step 3.2.5

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

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

Step 3.2.6

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

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

Step 3.2.7

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

$x - 1 + (x + 2) (1 + 0)$

Step 3.2.8

Simplify by adding terms.

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Step 3.2.8.1

Add $1$ and $0$ .

$x - 1 + (x + 2) \cdot 1$

Step 3.2.8.2

Multiply $x + 2$ by $1$ .

$x - 1 + x + 2$

Step 3.2.8.3

Add $x$ and $x$ .

$2 x - 1 + 2$

Step 3.2.8.4

Add $- 1$ and $2$ .

$2 x + 1$

Step 4

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

$y' = 2 x + 1$

Step 5

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

$\frac{d y}{d x} = 2 x + 1$