Calculus Examples

$3 x^{4} y - 2 x^{3} y - 6 x + 8 y$

Step 1

By the Sum Rule, the derivative of $3 x^{4} y - 2 x^{3} y - 6 x + 8 y$ with respect to $x$ is $\frac{d}{d x} [3 x^{4} y] + \frac{d}{d x} [- 2 x^{3} y] + \frac{d}{d x} [- 6 x] + \frac{d}{d x} [8 y]$ .

$\frac{d}{d x} [3 x^{4} y] + \frac{d}{d x} [- 2 x^{3} y] + \frac{d}{d x} [- 6 x] + \frac{d}{d x} [8 y]$

Step 2

Evaluate $\frac{d}{d x} [3 x^{4} y]$ .

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Step 2.1

Since $3 y$ is constant with respect to $x$ , the derivative of $3 x^{4} y$ with respect to $x$ is $3 y \frac{d}{d x} [x^{4}]$ .

$3 y \frac{d}{d x} [x^{4}] + \frac{d}{d x} [- 2 x^{3} y] + \frac{d}{d x} [- 6 x] + \frac{d}{d x} [8 y]$

Step 2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

$3 y (4 x^{3}) + \frac{d}{d x} [- 2 x^{3} y] + \frac{d}{d x} [- 6 x] + \frac{d}{d x} [8 y]$

Step 2.3

Multiply $4$ by $3$ .

$12 y x^{3} + \frac{d}{d x} [- 2 x^{3} y] + \frac{d}{d x} [- 6 x] + \frac{d}{d x} [8 y]$

Step 3

Evaluate $\frac{d}{d x} [- 2 x^{3} y]$ .

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Step 3.1

Since $- 2 y$ is constant with respect to $x$ , the derivative of $- 2 x^{3} y$ with respect to $x$ is $- 2 y \frac{d}{d x} [x^{3}]$ .

$12 y x^{3} - 2 y \frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 6 x] + \frac{d}{d x} [8 y]$

Step 3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$12 y x^{3} - 2 y (3 x^{2}) + \frac{d}{d x} [- 6 x] + \frac{d}{d x} [8 y]$

Step 3.3

Multiply $3$ by $- 2$ .

$12 y x^{3} - 6 y x^{2} + \frac{d}{d x} [- 6 x] + \frac{d}{d x} [8 y]$

Step 4

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

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Step 4.1

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

$12 y x^{3} - 6 y x^{2} - 6 \frac{d}{d x} [x] + \frac{d}{d x} [8 y]$

Step 4.2

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

$12 y x^{3} - 6 y x^{2} - 6 \cdot 1 + \frac{d}{d x} [8 y]$

Step 4.3

Multiply $- 6$ by $1$ .

$12 y x^{3} - 6 y x^{2} - 6 + \frac{d}{d x} [8 y]$

Step 5

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

$12 y x^{3} - 6 y x^{2} - 6 + 0$

Step 6

Simplify.

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Step 6.1

Add $12 y x^{3} - 6 y x^{2} - 6$ and $0$ .

$12 y x^{3} - 6 y x^{2} - 6$

Step 6.2

Reorder terms.

$12 x^{3} y - 6 x^{2} y - 6$