Calculus Examples

$f (x) = 1 x y$

Step 1

Find the first derivative of the function.

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Step 1.1

Multiply $x$ by $1$ .

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

Step 1.2

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

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

Step 1.3

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

$y \cdot 1$

Step 1.4

Multiply $y$ by $1$ .

$y$

Step 2

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

$y = 0$

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

Multiply $x$ by $1$ .

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

Step 4.1.2

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

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

Step 4.1.3

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

$y \cdot 1$

Step 4.1.4

Multiply $y$ by $1$ .

$f' (x) = y$

Step 4.2

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

$y$

Step 5

Set the first derivative equal to $0$ .

$y = 0$

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