Calculus Examples

$x y^{2} - 5 y + 6$

Step 1

By the Sum Rule, the derivative of $x y^{2} - 5 y + 6$ with respect to $x$ is $\frac{d}{d x} [x y^{2}] + \frac{d}{d x} [- 5 y] + \frac{d}{d x} [6]$ .

$\frac{d}{d x} [x y^{2}] + \frac{d}{d x} [- 5 y] + \frac{d}{d x} [6]$

Step 2

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

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Step 2.1

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

$y^{2} \frac{d}{d x} [x] + \frac{d}{d x} [- 5 y] + \frac{d}{d x} [6]$

Step 2.2

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

$y^{2} \cdot 1 + \frac{d}{d x} [- 5 y] + \frac{d}{d x} [6]$

Step 2.3

Multiply $y^{2}$ by $1$ .

$y^{2} + \frac{d}{d x} [- 5 y] + \frac{d}{d x} [6]$

Step 3

Differentiate using the Constant Rule.

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Step 3.1

Since $- 5 y$ is constant with respect to $x$ , the derivative of $- 5 y$ with respect to $x$ is $0$ .

$y^{2} + 0 + \frac{d}{d x} [6]$

Step 3.2

Since $6$ is constant with respect to $x$ , the derivative of $6$ with respect to $x$ is $0$ .

$y^{2} + 0 + 0$

Step 4

Combine terms.

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Step 4.1

Add $y^{2}$ and $0$ .

$y^{2} + 0$

Step 4.2

Add $y^{2}$ and $0$ .

$y^{2}$