Calculus Examples

$f (x) = 2 p x^{2} + (2 p x (\frac{5 l}{p r^{2}}))$

Step 1

By the Sum Rule, the derivative of $2 p x^{2} + (2 p x (\frac{5 l}{p r^{2}}))$ with respect to $x$ is $\frac{d}{d x} [2 p x^{2}] + \frac{d}{d x} [2 p x (\frac{5 l}{p r^{2}})]$ .

$\frac{d}{d x} [2 p x^{2}] + \frac{d}{d x} [2 p x (\frac{5 l}{p r^{2}})]$

Step 2

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

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Step 2.1

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

$2 p \frac{d}{d x} [x^{2}] + \frac{d}{d x} [2 p x (\frac{5 l}{p r^{2}})]$

Step 2.2

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

$2 p (2 x) + \frac{d}{d x} [2 p x (\frac{5 l}{p r^{2}})]$

Step 2.3

Multiply $2$ by $2$ .

$4 p x + \frac{d}{d x} [2 p x (\frac{5 l}{p r^{2}})]$

Step 3

Evaluate $\frac{d}{d x} [2 p x (\frac{5 l}{p r^{2}})]$ .

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Step 3.1

Combine $2$ and $\frac{5 l}{p r^{2}}$ .

$4 p x + \frac{d}{d x} [p x \frac{2 (5 l)}{p r^{2}}]$

Step 3.2

Multiply $5$ by $2$ .

$4 p x + \frac{d}{d x} [p x \frac{10 l}{p r^{2}}]$

Step 3.3

Combine $p$ and $\frac{10 l}{p r^{2}}$ .

$4 p x + \frac{d}{d x} [x \frac{p (10 l)}{p r^{2}}]$

Step 3.4

Combine $x$ and $\frac{p (10 l)}{p r^{2}}$ .

$4 p x + \frac{d}{d x} [\frac{x (p (10 l))}{p r^{2}}]$

Step 3.5

Move $10$ to the left of $p$ .

$4 p x + \frac{d}{d x} [\frac{x (10 \cdot p l)}{p r^{2}}]$

Step 3.6

Move $10$ to the left of $x$ .

$4 p x + \frac{d}{d x} [\frac{10 \cdot x p l}{p r^{2}}]$

Step 3.7

Cancel the common factor of $p$ .

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Step 3.7.1

Cancel the common factor.

$4 p x + \frac{d}{d x} [\frac{10 x p l}{p r^{2}}]$

Step 3.7.2

Rewrite the expression.

$4 p x + \frac{d}{d x} [\frac{10 x l}{r^{2}}]$

Step 3.8

Since $\frac{10 l}{r^{2}}$ is constant with respect to $x$ , the derivative of $\frac{10 x l}{r^{2}}$ with respect to $x$ is $\frac{10 l}{r^{2}} \frac{d}{d x} [x]$ .

$4 p x + \frac{10 l}{r^{2}} \frac{d}{d x} [x]$

Step 3.9

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

$4 p x + \frac{10 l}{r^{2}} \cdot 1$

Step 3.10

Multiply $\frac{10 l}{r^{2}}$ by $1$ .

$4 p x + \frac{10 l}{r^{2}}$