Calculus Examples

$f (x) = 2 x y$

Step 1

Find the first derivative of the function.

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Step 1.1

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

$2 y \frac{d}{d x} [x]$

Step 1.2

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

$2 y \cdot 1$

Step 1.3

Multiply $2$ by $1$ .

$2 y$

Step 2

Since $2 y$ is constant with respect to $x$ , the derivative of $2 y$ with respect to $x$ is $0$ .

$f'' (x) = 0$

Step 3

To find the local maximum and minimum values of the function, set the derivative equal to $0$ and solve.

$2 y = 0$

Step 4

Find the first derivative.

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Step 4.1

Find the first derivative.

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Step 4.1.1

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

$2 y \frac{d}{d x} [x]$

Step 4.1.2

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

$2 y \cdot 1$

Step 4.1.3

Multiply $2$ by $1$ .

$f' (x) = 2 y$

Step 4.2

The first derivative of $f (x)$ with respect to $x$ is $2 y$ .

$2 y$

Step 5

Set the first derivative equal to $0$ then solve the equation $2 y = 0$ .

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Step 5.1

Set the first derivative equal to $0$ .

$2 y = 0$

Step 5.2

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

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Step 5.2.1

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

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

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Cancel the common factor of $2$ .

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Step 5.2.2.1.1

Cancel the common factor.

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

Step 5.2.2.1.2

Divide $y$ by $1$ .

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

Step 5.2.3

Simplify the right side.

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Step 5.2.3.1

Divide $0$ by $2$ .

$y = 0$

Step 6

Critical points to evaluate.

$x = 0$

Step 7

Evaluate the second derivative at $x = 0$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$0$

Step 8

Since the first derivative test failed, there are no local extrema.

No Local Extrema

Step 9