Calculus Examples

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

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

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

Find the derivative.

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

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

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

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

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

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

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

Multiply $2$ by $5$ .

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

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

$10 x + 0$

Add $10 x$ and $0$ .

$10 x$

Divide each term by $10$ and simplify.

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

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

Reduce the expression by cancelling the common factors.

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Cancel the common factor.

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

Divide $x$ by $1$ .

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

Divide $0$ by $10$ .

$x = 0$

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

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

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

Simplify the result.

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Simplify each term.

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Remove parentheses.

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

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

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

Multiply $5$ by $0$ .

$f (0) = 0 + 5$

Add $0$ and $5$ .

$f (0) = 5$

The final answer is $5$ .

$5$

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

$y = 5$

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