Calculus Examples

$x = \frac{500}{\ln (p^{2} + 1)}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d p} (x) = \frac{d}{d p} (\frac{500}{\ln (p^{2} + 1)})$

Step 2

The derivative of $x$ with respect to $p$ is $x'$ .

$x'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

Differentiate using the Constant Multiple Rule.

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Step 3.1.1

Since $500$ is constant with respect to $p$ , the derivative of $\frac{500}{\ln (p^{2} + 1)}$ with respect to $p$ is $500 \frac{d}{d p} [\frac{1}{\ln (p^{2} + 1)}]$ .

$500 \frac{d}{d p} [\frac{1}{\ln (p^{2} + 1)}]$

Step 3.1.2

Rewrite $\frac{1}{\ln (p^{2} + 1)}$ as $\ln^{- 1} (p^{2} + 1)$ .

$500 \frac{d}{d p} [\ln^{- 1} (p^{2} + 1)]$

Step 3.2

Differentiate using the chain rule, which states that $\frac{d}{d p} [f (g (p))]$ is $f' (g (p)) g' (p)$ where $f (p) = p^{- 1}$ and $g (p) = \ln (p^{2} + 1)$ .

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Step 3.2.1

To apply the Chain Rule, set $u_{1}$ as $\ln (p^{2} + 1)$ .

$500 (\frac{d}{d u_{1}} [{u_{1}}^{- 1}] \frac{d}{d p} [\ln (p^{2} + 1)])$

Step 3.2.2

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

$500 (- {u_{1}}^{- 2} \frac{d}{d p} [\ln (p^{2} + 1)])$

Step 3.2.3

Replace all occurrences of $u_{1}$ with $\ln (p^{2} + 1)$ .

$500 (- \ln^{- 2} (p^{2} + 1) \frac{d}{d p} [\ln (p^{2} + 1)])$

Step 3.3

Multiply $- 1$ by $500$ .

$- 500 (\ln^{- 2} (p^{2} + 1) \frac{d}{d p} [\ln (p^{2} + 1)])$

Step 3.4

Differentiate using the chain rule, which states that $\frac{d}{d p} [f (g (p))]$ is $f' (g (p)) g' (p)$ where $f (p) = \ln (p)$ and $g (p) = p^{2} + 1$ .

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Step 3.4.1

To apply the Chain Rule, set $u_{2}$ as $p^{2} + 1$ .

$- 500 \ln^{- 2} (p^{2} + 1) (\frac{d}{d u_{2}} [\ln (u_{2})] \frac{d}{d p} [p^{2} + 1])$

Step 3.4.2

The derivative of $\ln (u_{2})$ with respect to $u_{2}$ is $\frac{1}{u_{2}}$ .

$- 500 \ln^{- 2} (p^{2} + 1) (\frac{1}{u_{2}} \frac{d}{d p} [p^{2} + 1])$

Step 3.4.3

Replace all occurrences of $u_{2}$ with $p^{2} + 1$ .

$- 500 \ln^{- 2} (p^{2} + 1) (\frac{1}{p^{2} + 1} \frac{d}{d p} [p^{2} + 1])$

Step 3.5

Differentiate.

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Step 3.5.1

Combine $\frac{1}{p^{2} + 1}$ and $- 500$ .

$\frac{- 500}{p^{2} + 1} \ln^{- 2} (p^{2} + 1) \frac{d}{d p} [p^{2} + 1]$

Step 3.5.2

Combine fractions.

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Step 3.5.2.1

Combine $\frac{- 500}{p^{2} + 1}$ and $\ln^{- 2} (p^{2} + 1)$ .

$\frac{- 500 \ln^{- 2} (p^{2} + 1)}{p^{2} + 1} \frac{d}{d p} [p^{2} + 1]$

Step 3.5.2.2

Simplify the expression.

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Step 3.5.2.2.1

Move $\ln^{- 2} (p^{2} + 1)$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{- 500}{(p^{2} + 1) \ln^{2} (p^{2} + 1)} \frac{d}{d p} [p^{2} + 1]$

Step 3.5.2.2.2

Move the negative in front of the fraction.

$- \frac{500}{(p^{2} + 1) \ln^{2} (p^{2} + 1)} \frac{d}{d p} [p^{2} + 1]$

Step 3.5.3

By the Sum Rule, the derivative of $p^{2} + 1$ with respect to $p$ is $\frac{d}{d p} [p^{2}] + \frac{d}{d p} [1]$ .

$- \frac{500}{(p^{2} + 1) \ln^{2} (p^{2} + 1)} (\frac{d}{d p} [p^{2}] + \frac{d}{d p} [1])$

Step 3.5.4

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

$- \frac{500}{(p^{2} + 1) \ln^{2} (p^{2} + 1)} (2 p + \frac{d}{d p} [1])$

Step 3.5.5

Since $1$ is constant with respect to $p$ , the derivative of $1$ with respect to $p$ is $0$ .

$- \frac{500}{(p^{2} + 1) \ln^{2} (p^{2} + 1)} (2 p + 0)$

Step 3.5.6

Combine fractions.

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Step 3.5.6.1

Add $2 p$ and $0$ .

$- \frac{500}{(p^{2} + 1) \ln^{2} (p^{2} + 1)} (2 p)$

Step 3.5.6.2

Multiply $2$ by $- 1$ .

$- 2 \frac{500}{(p^{2} + 1) \ln^{2} (p^{2} + 1)} p$

Step 3.5.6.3

Combine $- 2$ and $\frac{500}{(p^{2} + 1) \ln^{2} (p^{2} + 1)}$ .

$\frac{- 2 \cdot 500}{(p^{2} + 1) \ln^{2} (p^{2} + 1)} p$

Step 3.5.6.4

Multiply $- 2$ by $500$ .

$\frac{- 1000}{(p^{2} + 1) \ln^{2} (p^{2} + 1)} p$

Step 3.5.6.5

Combine $\frac{- 1000}{(p^{2} + 1) \ln^{2} (p^{2} + 1)}$ and $p$ .

$\frac{- 1000 p}{(p^{2} + 1) \ln^{2} (p^{2} + 1)}$

Step 3.5.6.6

Simplify the expression.

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Step 3.5.6.6.1

Move the negative in front of the fraction.

$- \frac{1000 p}{(p^{2} + 1) \ln^{2} (p^{2} + 1)}$

Step 3.5.6.6.2

Reorder factors in $- \frac{1000 p}{(p^{2} + 1) \ln^{2} (p^{2} + 1)}$ .

$- \frac{1000 p}{\ln^{2} (p^{2} + 1) (p^{2} + 1)}$

Step 4

Reform the equation by setting the left side equal to the right side.

$x' = - \frac{1000 p}{\ln^{2} (p^{2} + 1) (p^{2} + 1)}$

Step 5

Replace $x'$ with $\frac{d x}{d p}$ .

$\frac{d x}{d p} = - \frac{1000 p}{\ln^{2} (p^{2} + 1) (p^{2} + 1)}$