Calculus Examples

$5 x^{2} - 2 x y + 7 y^{2}$

Step 1

Differentiate.

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Step 1.1

By the Sum Rule, the derivative of $5 x^{2} - 2 x y + 7 y^{2}$ with respect to $y$ is $\frac{d}{d y} [5 x^{2}] + \frac{d}{d y} [- 2 x y] + \frac{d}{d y} [7 y^{2}]$ .

$\frac{d}{d y} [5 x^{2}] + \frac{d}{d y} [- 2 x y] + \frac{d}{d y} [7 y^{2}]$

Step 1.2

Since $5 x^{2}$ is constant with respect to $y$ , the derivative of $5 x^{2}$ with respect to $y$ is $0$ .

$0 + \frac{d}{d y} [- 2 x y] + \frac{d}{d y} [7 y^{2}]$

Step 2

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

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Step 2.1

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

$0 - 2 x \frac{d}{d y} [y] + \frac{d}{d y} [7 y^{2}]$

Step 2.2

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

$0 - 2 x \cdot 1 + \frac{d}{d y} [7 y^{2}]$

Step 2.3

Multiply $- 2$ by $1$ .

$0 - 2 x + \frac{d}{d y} [7 y^{2}]$

Step 3

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

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Step 3.1

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

$0 - 2 x + 7 \frac{d}{d y} [y^{2}]$

Step 3.2

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

$0 - 2 x + 7 (2 y)$

Step 3.3

Multiply $2$ by $7$ .

$0 - 2 x + 14 y$

Step 4

Subtract $2 x$ from $0$ .

$- 2 x + 14 y$