Calculus Examples

$y = 5 x^{2} + 5$

Step 1

Set $y$ as a function of $x$ .

$f (x) = 5 x^{2} + 5$

Step 2

Find the derivative.

Tap for more steps...

Step 2.1

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

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

Step 2.2

Evaluate $\frac{d}{d x} [5 x^{2}]$ .

Tap for more steps...

Step 2.2.1

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

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

Step 2.2.2

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

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

Step 2.2.3

Multiply $2$ by $5$ .

$10 x + \frac{d}{d x} [5]$

Step 2.3

Differentiate using the Constant Rule.

Tap for more steps...

Step 2.3.1

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

$10 x + 0$

Step 2.3.2

Add $10 x$ and $0$ .

$10 x$

Step 3

Divide each term in $10 x = 0$ by $10$ and simplify.

Tap for more steps...

Step 3.1

Divide each term in $10 x = 0$ by $10$ .

$\frac{10 x}{10} = \frac{0}{10}$

Step 3.2

Simplify the left side.

Tap for more steps...

Step 3.2.1

Cancel the common factor of $10$ .

Tap for more steps...

Step 3.2.1.1

Cancel the common factor.

$\frac{10 x}{10} = \frac{0}{10}$

Step 3.2.1.2

Divide $x$ by $1$ .

$x = \frac{0}{10}$

Step 3.3

Simplify the right side.

Tap for more steps...

Step 3.3.1

Divide $0$ by $10$ .

$x = 0$

Step 4

Solve the original function $f (x) = 5 x^{2} + 5$ at $x = 0$ .

Tap for more steps...

Step 4.1

Replace the variable $x$ with $0$ in the expression.

$f (0) = 5 {(0)}^{2} + 5$

Step 4.2

Simplify the result.

Tap for more steps...

Step 4.2.1

Simplify each term.

Tap for more steps...

Step 4.2.1.1

Raising $0$ to any positive power yields $0$ .

$f (0) = 5 \cdot 0 + 5$

Step 4.2.1.2

Multiply $5$ by $0$ .

$f (0) = 0 + 5$

Step 4.2.2

Add $0$ and $5$ .

$f (0) = 5$

Step 4.2.3

The final answer is $5$ .

$5$

Step 5

The horizontal tangent line on function $f (x) = 5 x^{2} + 5$ is $y = 5$ .

$y = 5$

Step 6

Enter YOUR Problem