Calculus Examples

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

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (2 x^{2} + 5)$

Step 2

The derivative of $y$ with respect to $x$ is $y'$ .

$y'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

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

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

Step 3.2

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

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Step 3.2.1

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

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

Step 3.2.2

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

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

Step 3.2.3

Multiply $2$ by $2$ .

$4 x + \frac{d}{d x} [5]$

Step 3.3

Differentiate using the Constant Rule.

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Step 3.3.1

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

$4 x + 0$

Step 3.3.2

Add $4 x$ and $0$ .

$4 x$

Step 4

Reform the equation by setting the left side equal to the right side.

$y' = 4 x$

Step 5

Replace $y'$ with $\frac{d y}{d x}$ .

$\frac{d y}{d x} = 4 x$

Step 6

Divide each term in $4 x = 0$ by $4$ and simplify.

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Step 6.1

Divide each term in $4 x = 0$ by $4$ .

$\frac{4 x}{4} = \frac{0}{4}$

Step 6.2

Simplify the left side.

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Step 6.2.1

Cancel the common factor of $4$ .

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Step 6.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{0}{4}$

Step 6.2.1.2

Divide $x$ by $1$ .

$x = \frac{0}{4}$

Step 6.3

Simplify the right side.

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Step 6.3.1

Divide $0$ by $4$ .

$x = 0$

Step 7

Solve for $y$ .

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Step 7.1

Remove parentheses.

$y = 2 \cdot 0^{2} + 5$

Step 7.2

Simplify $2 \cdot 0^{2} + 5$ .

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Step 7.2.1

Simplify each term.

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Step 7.2.1.1

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

$y = 2 \cdot 0 + 5$

Step 7.2.1.2

Multiply $2$ by $0$ .

$y = 0 + 5$

Step 7.2.2

Add $0$ and $5$ .

$y = 5$

Step 8

Find the points where $\frac{d y}{d x} = 0$ .

$(0, 5)$

Step 9