Calculus Examples

$f (x) = 7 x^{2} + 7 y^{2} + 7 x y - 168 x + 72$

Step 1

By the Sum Rule, the derivative of $7 x^{2} + 7 y^{2} + 7 x y - 168 x + 72$ with respect to $x$ is $\frac{d}{d x} [7 x^{2}] + \frac{d}{d x} [7 y^{2}] + \frac{d}{d x} [7 x y] + \frac{d}{d x} [- 168 x] + \frac{d}{d x} [72]$ .

$\frac{d}{d x} [7 x^{2}] + \frac{d}{d x} [7 y^{2}] + \frac{d}{d x} [7 x y] + \frac{d}{d x} [- 168 x] + \frac{d}{d x} [72]$

Step 2

Evaluate $\frac{d}{d x} [7 x^{2}]$ .

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Step 2.1

Since $7$ is constant with respect to $x$ , the derivative of $7 x^{2}$ with respect to $x$ is $7 \frac{d}{d x} [x^{2}]$ .

$7 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [7 y^{2}] + \frac{d}{d x} [7 x y] + \frac{d}{d x} [- 168 x] + \frac{d}{d x} [72]$

Step 2.2

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

$7 (2 x) + \frac{d}{d x} [7 y^{2}] + \frac{d}{d x} [7 x y] + \frac{d}{d x} [- 168 x] + \frac{d}{d x} [72]$

Step 2.3

Multiply $2$ by $7$ .

$14 x + \frac{d}{d x} [7 y^{2}] + \frac{d}{d x} [7 x y] + \frac{d}{d x} [- 168 x] + \frac{d}{d x} [72]$

Step 3

Since $7 y^{2}$ is constant with respect to $x$ , the derivative of $7 y^{2}$ with respect to $x$ is $0$ .

$14 x + 0 + \frac{d}{d x} [7 x y] + \frac{d}{d x} [- 168 x] + \frac{d}{d x} [72]$

Step 4

Evaluate $\frac{d}{d x} [7 x y]$ .

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Step 4.1

Since $7 y$ is constant with respect to $x$ , the derivative of $7 x y$ with respect to $x$ is $7 y \frac{d}{d x} [x]$ .

$14 x + 0 + 7 y \frac{d}{d x} [x] + \frac{d}{d x} [- 168 x] + \frac{d}{d x} [72]$

Step 4.2

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

$14 x + 0 + 7 y \cdot 1 + \frac{d}{d x} [- 168 x] + \frac{d}{d x} [72]$

Step 4.3

Multiply $7$ by $1$ .

$14 x + 0 + 7 y + \frac{d}{d x} [- 168 x] + \frac{d}{d x} [72]$

Step 5

Evaluate $\frac{d}{d x} [- 168 x]$ .

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Step 5.1

Since $- 168$ is constant with respect to $x$ , the derivative of $- 168 x$ with respect to $x$ is $- 168 \frac{d}{d x} [x]$ .

$14 x + 0 + 7 y - 168 \frac{d}{d x} [x] + \frac{d}{d x} [72]$

Step 5.2

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

$14 x + 0 + 7 y - 168 \cdot 1 + \frac{d}{d x} [72]$

Step 5.3

Multiply $- 168$ by $1$ .

$14 x + 0 + 7 y - 168 + \frac{d}{d x} [72]$

Step 6

Since $72$ is constant with respect to $x$ , the derivative of $72$ with respect to $x$ is $0$ .

$14 x + 0 + 7 y - 168 + 0$

Step 7

Combine terms.

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Step 7.1

Add $14 x$ and $0$ .

$14 x + 7 y - 168 + 0$

Step 7.2

Add $14 x + 7 y - 168$ and $0$ .

$14 x + 7 y - 168$