Calculus Examples

$y = x^{2} + 1$

Step 1

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

$f (x) = x^{2} + 1$

Step 2

Find the derivative.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

$2 x + 0$

Step 2.4

Add $2 x$ and $0$ .

$2 x$

Step 3

Divide each term in $2 x = 0$ by $2$ and simplify.

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Step 3.1

Divide each term in $2 x = 0$ by $2$ .

$\frac{2 x}{2} = \frac{0}{2}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $2$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{0}{2}$

Step 3.2.1.2

Divide $x$ by $1$ .

$x = \frac{0}{2}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Divide $0$ by $2$ .

$x = 0$

Step 4

Solve the original function $f (x) = x^{2} + 1$ at $x = 0$ .

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Step 4.1

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

$f (0) = {(0)}^{2} + 1$

Step 4.2

Simplify the result.

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Step 4.2.1

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

$f (0) = 0 + 1$

Step 4.2.2

Add $0$ and $1$ .

$f (0) = 1$

Step 4.2.3

The final answer is $1$ .

$1$

Step 5

The horizontal tangent line on function $f (x) = x^{2} + 1$ is $y = 1$ .

$y = 1$

Step 6