Calculus Examples

$2 x^{2} + 3 x y + 4 y^{2} - 2 x + 10 y$

Step 1

By the Sum Rule, the derivative of $2 x^{2} + 3 x y + 4 y^{2} - 2 x + 10 y$ with respect to $x$ is $\frac{d}{d x} [2 x^{2}] + \frac{d}{d x} [3 x y] + \frac{d}{d x} [4 y^{2}] + \frac{d}{d x} [- 2 x] + \frac{d}{d x} [10 y]$ .

$\frac{d}{d x} [2 x^{2}] + \frac{d}{d x} [3 x y] + \frac{d}{d x} [4 y^{2}] + \frac{d}{d x} [- 2 x] + \frac{d}{d x} [10 y]$

Step 2

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

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Step 2.1

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

$2 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [3 x y] + \frac{d}{d x} [4 y^{2}] + \frac{d}{d x} [- 2 x] + \frac{d}{d x} [10 y]$

Step 2.2

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

$2 (2 x) + \frac{d}{d x} [3 x y] + \frac{d}{d x} [4 y^{2}] + \frac{d}{d x} [- 2 x] + \frac{d}{d x} [10 y]$

Step 2.3

Multiply $2$ by $2$ .

$4 x + \frac{d}{d x} [3 x y] + \frac{d}{d x} [4 y^{2}] + \frac{d}{d x} [- 2 x] + \frac{d}{d x} [10 y]$

Step 3

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

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Step 3.1

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

$4 x + 3 y \frac{d}{d x} [x] + \frac{d}{d x} [4 y^{2}] + \frac{d}{d x} [- 2 x] + \frac{d}{d x} [10 y]$

Step 3.2

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

$4 x + 3 y \cdot 1 + \frac{d}{d x} [4 y^{2}] + \frac{d}{d x} [- 2 x] + \frac{d}{d x} [10 y]$

Step 3.3

Multiply $3$ by $1$ .

$4 x + 3 y + \frac{d}{d x} [4 y^{2}] + \frac{d}{d x} [- 2 x] + \frac{d}{d x} [10 y]$

Step 4

Since $4 y^{2}$ is constant with respect to $x$ , the derivative of $4 y^{2}$ with respect to $x$ is $0$ .

$4 x + 3 y + 0 + \frac{d}{d x} [- 2 x] + \frac{d}{d x} [10 y]$

Step 5

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

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Step 5.1

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

$4 x + 3 y + 0 - 2 \frac{d}{d x} [x] + \frac{d}{d x} [10 y]$

Step 5.2

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

$4 x + 3 y + 0 - 2 \cdot 1 + \frac{d}{d x} [10 y]$

Step 5.3

Multiply $- 2$ by $1$ .

$4 x + 3 y + 0 - 2 + \frac{d}{d x} [10 y]$

Step 6

Since $10 y$ is constant with respect to $x$ , the derivative of $10 y$ with respect to $x$ is $0$ .

$4 x + 3 y + 0 - 2 + 0$

Step 7

Combine terms.

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Step 7.1

Add $4 x + 3 y$ and $0$ .

$4 x + 3 y - 2 + 0$

Step 7.2

Add $4 x + 3 y - 2$ and $0$ .

$4 x + 3 y - 2$