Calculus Examples

$y = (x - 1) (x^{2} + 3 x - 5)$

Step 1

Find the first derivative.

Tap for more steps...

Step 1.1

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 $g (x) = x^{2} + 3 x - 5$ .

$(x - 1) \frac{d}{d x} [x^{2} + 3 x - 5] + (x^{2} + 3 x - 5) \frac{d}{d x} [x - 1]$

Step 1.2

Differentiate.

Tap for more steps...

Step 1.2.1

By the Sum Rule, the derivative of $x^{2} + 3 x - 5$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [3 x] + \frac{d}{d x} [- 5]$ .

$(x - 1) (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [3 x] + \frac{d}{d x} [- 5]) + (x^{2} + 3 x - 5) \frac{d}{d x} [x - 1]$

Step 1.2.2

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

$(x - 1) (2 x + \frac{d}{d x} [3 x] + \frac{d}{d x} [- 5]) + (x^{2} + 3 x - 5) \frac{d}{d x} [x - 1]$

Step 1.2.3

Since $3$ is constant with respect to $x$ , the derivative of $3 x$ with respect to $x$ is $3 \frac{d}{d x} [x]$ .

$(x - 1) (2 x + 3 \frac{d}{d x} [x] + \frac{d}{d x} [- 5]) + (x^{2} + 3 x - 5) \frac{d}{d x} [x - 1]$

Step 1.2.4

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

$(x - 1) (2 x + 3 \cdot 1 + \frac{d}{d x} [- 5]) + (x^{2} + 3 x - 5) \frac{d}{d x} [x - 1]$

Step 1.2.5

Multiply $3$ by $1$ .

$(x - 1) (2 x + 3 + \frac{d}{d x} [- 5]) + (x^{2} + 3 x - 5) \frac{d}{d x} [x - 1]$

Step 1.2.6

Since $- 5$ is constant with respect to $x$ , the derivative of $- 5$ with respect to $x$ is $0$ .

$(x - 1) (2 x + 3 + 0) + (x^{2} + 3 x - 5) \frac{d}{d x} [x - 1]$

Step 1.2.7

Add $2 x + 3$ and $0$ .

$(x - 1) (2 x + 3) + (x^{2} + 3 x - 5) \frac{d}{d x} [x - 1]$

Step 1.2.8

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

$(x - 1) (2 x + 3) + (x^{2} + 3 x - 5) (\frac{d}{d x} [x] + \frac{d}{d x} [- 1])$

Step 1.2.9

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

$(x - 1) (2 x + 3) + (x^{2} + 3 x - 5) (1 + \frac{d}{d x} [- 1])$

Step 1.2.10

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

$(x - 1) (2 x + 3) + (x^{2} + 3 x - 5) (1 + 0)$

Step 1.2.11

Simplify the expression.

Tap for more steps...

Step 1.2.11.1

Add $1$ and $0$ .

$(x - 1) (2 x + 3) + (x^{2} + 3 x - 5) \cdot 1$

Step 1.2.11.2

Multiply $x^{2} + 3 x - 5$ by $1$ .

$(x - 1) (2 x + 3) + x^{2} + 3 x - 5$

Step 1.3

Simplify.

Tap for more steps...

Step 1.3.1

Apply the distributive property.

$(x - 1) (2 x) + (x - 1) \cdot 3 + x^{2} + 3 x - 5$

Step 1.3.2

Apply the distributive property.

$x (2 x) - 1 (2 x) + (x - 1) \cdot 3 + x^{2} + 3 x - 5$

Step 1.3.3

Apply the distributive property.

$x (2 x) - 1 (2 x) + x \cdot 3 - 1 \cdot 3 + x^{2} + 3 x - 5$

Step 1.3.4

Combine terms.

Tap for more steps...

Step 1.3.4.1

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

$2 (x^{1} x) - 1 (2 x) + x \cdot 3 - 1 \cdot 3 + x^{2} + 3 x - 5$

Step 1.3.4.2

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

$2 (x^{1} x^{1}) - 1 (2 x) + x \cdot 3 - 1 \cdot 3 + x^{2} + 3 x - 5$

Step 1.3.4.3

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

$2 x^{1 + 1} - 1 (2 x) + x \cdot 3 - 1 \cdot 3 + x^{2} + 3 x - 5$

Step 1.3.4.4

Add $1$ and $1$ .

$2 x^{2} - 1 (2 x) + x \cdot 3 - 1 \cdot 3 + x^{2} + 3 x - 5$

Step 1.3.4.5

Multiply $2$ by $- 1$ .

$2 x^{2} - 2 x + x \cdot 3 - 1 \cdot 3 + x^{2} + 3 x - 5$

Step 1.3.4.6

Move $3$ to the left of $x$ .

$2 x^{2} - 2 x + 3 \cdot x - 1 \cdot 3 + x^{2} + 3 x - 5$

Step 1.3.4.7

Multiply $- 1$ by $3$ .

$2 x^{2} - 2 x + 3 x - 3 + x^{2} + 3 x - 5$

Step 1.3.4.8

Add $- 2 x$ and $3 x$ .

$2 x^{2} + x - 3 + x^{2} + 3 x - 5$

Step 1.3.4.9

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

$3 x^{2} + x - 3 + 3 x - 5$

Step 1.3.4.10

Add $x$ and $3 x$ .

$3 x^{2} + 4 x - 3 - 5$

Step 1.3.4.11

Subtract $5$ from $- 3$ .

$f' (x) = 3 x^{2} + 4 x - 8$

Step 2

Find the second derivative.

Tap for more steps...

Step 2.1

By the Sum Rule, the derivative of $3 x^{2} + 4 x - 8$ with respect to $x$ is $\frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [4 x] + \frac{d}{d x} [- 8]$ .

$\frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [4 x] + \frac{d}{d x} [- 8]$

Step 2.2

Evaluate $\frac{d}{d x} [3 x^{2}]$ .

Tap for more steps...

Step 2.2.1

Since $3$ is constant with respect to $x$ , the derivative of $3 x^{2}$ with respect to $x$ is $3 \frac{d}{d x} [x^{2}]$ .

$3 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [4 x] + \frac{d}{d x} [- 8]$

Step 2.2.2

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

$3 (2 x) + \frac{d}{d x} [4 x] + \frac{d}{d x} [- 8]$

Step 2.2.3

Multiply $2$ by $3$ .

$6 x + \frac{d}{d x} [4 x] + \frac{d}{d x} [- 8]$

Step 2.3

Evaluate $\frac{d}{d x} [4 x]$ .

Tap for more steps...

Step 2.3.1

Since $4$ is constant with respect to $x$ , the derivative of $4 x$ with respect to $x$ is $4 \frac{d}{d x} [x]$ .

$6 x + 4 \frac{d}{d x} [x] + \frac{d}{d x} [- 8]$

Step 2.3.2

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

$6 x + 4 \cdot 1 + \frac{d}{d x} [- 8]$

Step 2.3.3

Multiply $4$ by $1$ .

$6 x + 4 + \frac{d}{d x} [- 8]$

Step 2.4

Differentiate using the Constant Rule.

Tap for more steps...

Step 2.4.1

Since $- 8$ is constant with respect to $x$ , the derivative of $- 8$ with respect to $x$ is $0$ .

$6 x + 4 + 0$

Step 2.4.2

Add $6 x + 4$ and $0$ .

$f'' (x) = 6 x + 4$

Step 3

Find the third derivative.

Tap for more steps...

Step 3.1

By the Sum Rule, the derivative of $6 x + 4$ with respect to $x$ is $\frac{d}{d x} [6 x] + \frac{d}{d x} [4]$ .

$\frac{d}{d x} [6 x] + \frac{d}{d x} [4]$

Step 3.2

Evaluate $\frac{d}{d x} [6 x]$ .

Tap for more steps...

Step 3.2.1

Since $6$ is constant with respect to $x$ , the derivative of $6 x$ with respect to $x$ is $6 \frac{d}{d x} [x]$ .

$6 \frac{d}{d x} [x] + \frac{d}{d x} [4]$

Step 3.2.2

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

$6 \cdot 1 + \frac{d}{d x} [4]$

Step 3.2.3

Multiply $6$ by $1$ .

$6 + \frac{d}{d x} [4]$

Step 3.3

Differentiate using the Constant Rule.

Tap for more steps...

Step 3.3.1

Since $4$ is constant with respect to $x$ , the derivative of $4$ with respect to $x$ is $0$ .

$6 + 0$

Step 3.3.2

Add $6$ and $0$ .

$f''' (x) = 6$

Step 4

Since $6$ is constant with respect to $x$ , the derivative of $6$ with respect to $x$ is $0$ .

$f^{4} (x) = 0$