Calculus Examples

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

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

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

Step 2

Differentiate.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

Simplify the expression.

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Step 2.4.1

Add $1$ and $0$ .

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

Step 2.4.2

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

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

Step 2.5

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

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

Step 2.6

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

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

Step 2.7

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

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

Step 2.8

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

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

Step 2.9

Add $2 x + 1$ and $0$ .

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

Step 3

Simplify.

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Step 3.1

Apply the distributive property.

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

Step 3.2

Apply the distributive property.

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

Step 3.3

Apply the distributive property.

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

Step 3.4

Combine terms.

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Step 3.4.1

Raise $x$ to the power of $1$ .

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

Step 3.4.2

Raise $x$ to the power of $1$ .

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

Step 3.4.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 3.4.4

Add $1$ and $1$ .

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

Step 3.4.5

Multiply $2$ by $- 1$ .

$x^{2} + x + 1 + 2 x^{2} - 2 x + x \cdot 1 - 1 \cdot 1$

Step 3.4.6

Multiply $x$ by $1$ .

$x^{2} + x + 1 + 2 x^{2} - 2 x + x - 1 \cdot 1$

Step 3.4.7

Multiply $- 1$ by $1$ .

$x^{2} + x + 1 + 2 x^{2} - 2 x + x - 1$

Step 3.4.8

Add $- 2 x$ and $x$ .

$x^{2} + x + 1 + 2 x^{2} - x - 1$

Step 3.4.9

Add $x^{2}$ and $2 x^{2}$ .

$3 x^{2} + x + 1 - x - 1$

Step 3.4.10

Subtract $x$ from $x$ .

$3 x^{2} + 0 + 1 - 1$

Step 3.4.11

Add $3 x^{2}$ and $0$ .

$3 x^{2} + 1 - 1$

Step 3.4.12

Subtract $1$ from $1$ .

$3 x^{2} + 0$

Step 3.4.13

Add $3 x^{2}$ and $0$ .

$3 x^{2}$