Precalculus Examples

$z = 6 x + 7 y$

Step 1

Find the first derivative of the function.

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Step 1.1

By the Sum Rule, the derivative of $6 x + 7 y$ with respect to $x$ is $\frac{d}{d x} [6 x] + \frac{d}{d x} [7 y]$ .

$\frac{d}{d x} [6 x] + \frac{d}{d x} [7 y]$

Step 1.2

Evaluate $\frac{d}{d x} [6 x]$ .

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Step 1.2.1

Since $6$ is constant with respect to $x$ , the derivative of $6 x$ with respect to $x$ is $6 \frac{d}{d x} [x]$ .

$6 \frac{d}{d x} [x] + \frac{d}{d x} [7 y]$

Step 1.2.2

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

$6 \cdot 1 + \frac{d}{d x} [7 y]$

Step 1.2.3

Multiply $6$ by $1$ .

$6 + \frac{d}{d x} [7 y]$

Step 1.3

Differentiate using the Constant Rule.

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Step 1.3.1

Since $7 y$ is constant with respect to $x$ , the derivative of $7 y$ with respect to $x$ is $0$ .

$6 + 0$

Step 1.3.2

Add $6$ and $0$ .

$6$

Step 2

Since $6$ is constant with respect to $x$ , the derivative of $6$ 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.

$6 = 0$

Step 4

Since there is no value of $x$ that makes the first derivative equal to $0$ , there are no local extrema.

No Local Extrema

Step 5

No Local Extrema

Step 6