Calculus Examples

$y^{2} + 10 x y + 12 x - 8$

Step 1

Differentiate.

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Step 1.1

By the Sum Rule, the derivative of $y^{2} + 10 x y + 12 x - 8$ with respect to $x$ is $\frac{d}{d x} [y^{2}] + \frac{d}{d x} [10 x y] + \frac{d}{d x} [12 x] + \frac{d}{d x} [- 8]$ .

$\frac{d}{d x} [y^{2}] + \frac{d}{d x} [10 x y] + \frac{d}{d x} [12 x] + \frac{d}{d x} [- 8]$

Step 1.2

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

$0 + \frac{d}{d x} [10 x y] + \frac{d}{d x} [12 x] + \frac{d}{d x} [- 8]$

Step 2

Evaluate $\frac{d}{d x} [10 x y]$ .

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Step 2.1

Since $10 y$ is constant with respect to $x$ , the derivative of $10 x y$ with respect to $x$ is $10 y \frac{d}{d x} [x]$ .

$0 + 10 y \frac{d}{d x} [x] + \frac{d}{d x} [12 x] + \frac{d}{d x} [- 8]$

Step 2.2

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

$0 + 10 y \cdot 1 + \frac{d}{d x} [12 x] + \frac{d}{d x} [- 8]$

Step 2.3

Multiply $10$ by $1$ .

$0 + 10 y + \frac{d}{d x} [12 x] + \frac{d}{d x} [- 8]$

Step 3

Evaluate $\frac{d}{d x} [12 x]$ .

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Step 3.1

Since $12$ is constant with respect to $x$ , the derivative of $12 x$ with respect to $x$ is $12 \frac{d}{d x} [x]$ .

$0 + 10 y + 12 \frac{d}{d x} [x] + \frac{d}{d x} [- 8]$

Step 3.2

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

$0 + 10 y + 12 \cdot 1 + \frac{d}{d x} [- 8]$

Step 3.3

Multiply $12$ by $1$ .

$0 + 10 y + 12 + \frac{d}{d x} [- 8]$

Step 4

Since $- 8$ is constant with respect to $x$ , the derivative of $- 8$ with respect to $x$ is $0$ .

$0 + 10 y + 12 + 0$

Step 5

Combine terms.

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Step 5.1

Add $0$ and $10 y$ .

$10 y + 12 + 0$

Step 5.2

Add $10 y + 12$ and $0$ .

$10 y + 12$