Calculus Examples

$15 x^{2} - 12 x y^{2} + 10 y + 33$

Step 1

By the Sum Rule, the derivative of $15 x^{2} - 12 x y^{2} + 10 y + 33$ with respect to $x$ is $\frac{d}{d x} [15 x^{2}] + \frac{d}{d x} [- 12 x y^{2}] + \frac{d}{d x} [10 y] + \frac{d}{d x} [33]$ .

$\frac{d}{d x} [15 x^{2}] + \frac{d}{d x} [- 12 x y^{2}] + \frac{d}{d x} [10 y] + \frac{d}{d x} [33]$

Step 2

Evaluate $\frac{d}{d x} [15 x^{2}]$ .

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Step 2.1

Since $15$ is constant with respect to $x$ , the derivative of $15 x^{2}$ with respect to $x$ is $15 \frac{d}{d x} [x^{2}]$ .

$15 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 12 x y^{2}] + \frac{d}{d x} [10 y] + \frac{d}{d x} [33]$

Step 2.2

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

$15 (2 x) + \frac{d}{d x} [- 12 x y^{2}] + \frac{d}{d x} [10 y] + \frac{d}{d x} [33]$

Step 2.3

Multiply $2$ by $15$ .

$30 x + \frac{d}{d x} [- 12 x y^{2}] + \frac{d}{d x} [10 y] + \frac{d}{d x} [33]$

Step 3

Evaluate $\frac{d}{d x} [- 12 x y^{2}]$ .

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Step 3.1

Since $- 12 y^{2}$ is constant with respect to $x$ , the derivative of $- 12 x y^{2}$ with respect to $x$ is $- 12 y^{2} \frac{d}{d x} [x]$ .

$30 x - 12 y^{2} \frac{d}{d x} [x] + \frac{d}{d x} [10 y] + \frac{d}{d x} [33]$

Step 3.2

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

$30 x - 12 y^{2} \cdot 1 + \frac{d}{d x} [10 y] + \frac{d}{d x} [33]$

Step 3.3

Multiply $- 12$ by $1$ .

$30 x - 12 y^{2} + \frac{d}{d x} [10 y] + \frac{d}{d x} [33]$

Step 4

Differentiate using the Constant Rule.

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Step 4.1

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

$30 x - 12 y^{2} + 0 + \frac{d}{d x} [33]$

Step 4.2

Since $33$ is constant with respect to $x$ , the derivative of $33$ with respect to $x$ is $0$ .

$30 x - 12 y^{2} + 0 + 0$

Step 5

Simplify.

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Step 5.1

Combine terms.

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Step 5.1.1

Add $30 x - 12 y^{2}$ and $0$ .

$30 x - 12 y^{2} + 0$

Step 5.1.2

Add $30 x - 12 y^{2}$ and $0$ .

$30 x - 12 y^{2}$

Step 5.2

Reorder terms.

$- 12 y^{2} + 30 x$