Calculus Examples

$5 x^{2} (x + 47)$

Step 1

Find the first derivative.

Tap for more steps...

Step 1.1

Since $5$ is constant with respect to $x$ , the derivative of $5 x^{2} (x + 47)$ with respect to $x$ is $5 \frac{d}{d x} [x^{2} (x + 47)]$ .

$5 \frac{d}{d x} [x^{2} (x + 47)]$

Step 1.2

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 $f (x) = x^{2}$ and $g (x) = x + 47$ .

$5 (x^{2} \frac{d}{d x} [x + 47] + (x + 47) \frac{d}{d x} [x^{2}])$

Step 1.3

Differentiate.

Tap for more steps...

Step 1.3.1

By the Sum Rule, the derivative of $x + 47$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [47]$ .

$5 (x^{2} (\frac{d}{d x} [x] + \frac{d}{d x} [47]) + (x + 47) \frac{d}{d x} [x^{2}])$

Step 1.3.2

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

$5 (x^{2} (1 + \frac{d}{d x} [47]) + (x + 47) \frac{d}{d x} [x^{2}])$

Step 1.3.3

Since $47$ is constant with respect to $x$ , the derivative of $47$ with respect to $x$ is $0$ .

$5 (x^{2} (1 + 0) + (x + 47) \frac{d}{d x} [x^{2}])$

Step 1.3.4

Simplify the expression.

Tap for more steps...

Step 1.3.4.1

Add $1$ and $0$ .

$5 (x^{2} \cdot 1 + (x + 47) \frac{d}{d x} [x^{2}])$

Step 1.3.4.2

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

$5 (x^{2} + (x + 47) \frac{d}{d x} [x^{2}])$

Step 1.3.5

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

$5 (x^{2} + (x + 47) (2 x))$

Step 1.3.6

Move $2$ to the left of $x + 47$ .

$5 (x^{2} + 2 \cdot (x + 47) x)$

Step 1.4

Simplify.

Tap for more steps...

Step 1.4.1

Apply the distributive property.

$5 (x^{2} + (2 x + 2 \cdot 47) x)$

Step 1.4.2

Apply the distributive property.

$5 (x^{2} + 2 x \cdot x + 2 \cdot 47 x)$

Step 1.4.3

Apply the distributive property.

$5 x^{2} + 5 (2 x \cdot x) + 5 (2 \cdot 47 x)$

Step 1.4.4

Combine terms.

Tap for more steps...

Step 1.4.4.1

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

$5 x^{2} + 5 (2 (x^{1} x)) + 5 (2 \cdot 47 x)$

Step 1.4.4.2

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

$5 x^{2} + 5 (2 (x^{1} x^{1})) + 5 (2 \cdot 47 x)$

Step 1.4.4.3

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

$5 x^{2} + 5 (2 x^{1 + 1}) + 5 (2 \cdot 47 x)$

Step 1.4.4.4

Add $1$ and $1$ .

$5 x^{2} + 5 (2 x^{2}) + 5 (2 \cdot 47 x)$

Step 1.4.4.5

Multiply $2$ by $5$ .

$5 x^{2} + 10 x^{2} + 5 (2 \cdot 47 x)$

Step 1.4.4.6

Multiply $2$ by $47$ .

$5 x^{2} + 10 x^{2} + 5 (94 x)$

Step 1.4.4.7

Multiply $94$ by $5$ .

$5 x^{2} + 10 x^{2} + 470 x$

Step 1.4.4.8

Add $5 x^{2}$ and $10 x^{2}$ .

$f' (x) = 15 x^{2} + 470 x$

Step 2

Find the second derivative.

Tap for more steps...

Step 2.1

By the Sum Rule, the derivative of $15 x^{2} + 470 x$ with respect to $x$ is $\frac{d}{d x} [15 x^{2}] + \frac{d}{d x} [470 x]$ .

$\frac{d}{d x} [15 x^{2}] + \frac{d}{d x} [470 x]$

Step 2.2

Evaluate $\frac{d}{d x} [15 x^{2}]$ .

Tap for more steps...

Step 2.2.1

Since $15$ is constant with respect to $x$ , the derivative of $15 x^{2}$ with respect to $x$ is $15 \frac{d}{d x} [x^{2}]$ .

$15 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [470 x]$

Step 2.2.2

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

$15 (2 x) + \frac{d}{d x} [470 x]$

Step 2.2.3

Multiply $2$ by $15$ .

$30 x + \frac{d}{d x} [470 x]$

Step 2.3

Evaluate $\frac{d}{d x} [470 x]$ .

Tap for more steps...

Step 2.3.1

Since $470$ is constant with respect to $x$ , the derivative of $470 x$ with respect to $x$ is $470 \frac{d}{d x} [x]$ .

$30 x + 470 \frac{d}{d x} [x]$

Step 2.3.2

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

$30 x + 470 \cdot 1$

Step 2.3.3

Multiply $470$ by $1$ .

$f'' (x) = 30 x + 470$

Step 3

Find the third derivative.

Tap for more steps...

Step 3.1

By the Sum Rule, the derivative of $30 x + 470$ with respect to $x$ is $\frac{d}{d x} [30 x] + \frac{d}{d x} [470]$ .

$\frac{d}{d x} [30 x] + \frac{d}{d x} [470]$

Step 3.2

Evaluate $\frac{d}{d x} [30 x]$ .

Tap for more steps...

Step 3.2.1

Since $30$ is constant with respect to $x$ , the derivative of $30 x$ with respect to $x$ is $30 \frac{d}{d x} [x]$ .

$30 \frac{d}{d x} [x] + \frac{d}{d x} [470]$

Step 3.2.2

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

$30 \cdot 1 + \frac{d}{d x} [470]$

Step 3.2.3

Multiply $30$ by $1$ .

$30 + \frac{d}{d x} [470]$

Step 3.3

Differentiate using the Constant Rule.

Tap for more steps...

Step 3.3.1

Since $470$ is constant with respect to $x$ , the derivative of $470$ with respect to $x$ is $0$ .

$30 + 0$

Step 3.3.2

Add $30$ and $0$ .

$f''' (x) = 30$