Calculus Examples

$p = \frac{144 r}{{(r + 0.6)}^{2}}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d r} (p) = \frac{d}{d r} (\frac{144 r}{{(r + 0.6)}^{2}})$

Step 2

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

$p'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

Since $144$ is constant with respect to $r$ , the derivative of $\frac{144 r}{{(r + 0.6)}^{2}}$ with respect to $r$ is $144 \frac{d}{d r} [\frac{r}{{(r + 0.6)}^{2}}]$ .

$144 \frac{d}{d r} [\frac{r}{{(r + 0.6)}^{2}}]$

Step 3.2

Differentiate using the Quotient Rule which states that $\frac{d}{d r} [\frac{f (r)}{g (r)}]$ is $\frac{g (r) \frac{d}{d r} [f (r)] - f (r) \frac{d}{d r} [g (r)]}{{g (r)}^{2}}$ where $f (r) = r$ and $g (r) = {(r + 0.6)}^{2}$ .

$144 \frac{{(r + 0.6)}^{2} \frac{d}{d r} [r] - r \frac{d}{d r} [{(r + 0.6)}^{2}]}{{({(r + 0.6)}^{2})}^{2}}$

Step 3.3

Differentiate using the Power Rule.

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Step 3.3.1

Multiply the exponents in ${({(r + 0.6)}^{2})}^{2}$ .

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Step 3.3.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$144 \frac{{(r + 0.6)}^{2} \frac{d}{d r} [r] - r \frac{d}{d r} [{(r + 0.6)}^{2}]}{{(r + 0.6)}^{2 \cdot 2}}$

Step 3.3.1.2

Multiply $2$ by $2$ .

$144 \frac{{(r + 0.6)}^{2} \frac{d}{d r} [r] - r \frac{d}{d r} [{(r + 0.6)}^{2}]}{{(r + 0.6)}^{4}}$

Step 3.3.2

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

$144 \frac{{(r + 0.6)}^{2} \cdot 1 - r \frac{d}{d r} [{(r + 0.6)}^{2}]}{{(r + 0.6)}^{4}}$

Step 3.3.3

Multiply ${(r + 0.6)}^{2}$ by $1$ .

$144 \frac{{(r + 0.6)}^{2} - r \frac{d}{d r} [{(r + 0.6)}^{2}]}{{(r + 0.6)}^{4}}$

Step 3.4

Differentiate using the chain rule, which states that $\frac{d}{d r} [f (g (r))]$ is $f' (g (r)) g' (r)$ where $f (r) = r^{2}$ and $g (r) = r + 0.6$ .

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Step 3.4.1

To apply the Chain Rule, set $u$ as $r + 0.6$ .

$144 \frac{{(r + 0.6)}^{2} - r (\frac{d}{d u} [u^{2}] \frac{d}{d r} [r + 0.6])}{{(r + 0.6)}^{4}}$

Step 3.4.2

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

$144 \frac{{(r + 0.6)}^{2} - r (2 u \frac{d}{d r} [r + 0.6])}{{(r + 0.6)}^{4}}$

Step 3.4.3

Replace all occurrences of $u$ with $r + 0.6$ .

$144 \frac{{(r + 0.6)}^{2} - r (2 (r + 0.6) \frac{d}{d r} [r + 0.6])}{{(r + 0.6)}^{4}}$

Step 3.5

Simplify with factoring out.

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Step 3.5.1

Multiply $2$ by $- 1$ .

$144 \frac{{(r + 0.6)}^{2} - 2 r ((r + 0.6) \frac{d}{d r} [r + 0.6])}{{(r + 0.6)}^{4}}$

Step 3.5.2

Factor $r + 0.6$ out of ${(r + 0.6)}^{2} - 2 r ((r + 0.6) \frac{d}{d r} [r + 0.6])$ .

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Step 3.5.2.1

Factor $r + 0.6$ out of ${(r + 0.6)}^{2}$ .

$144 \frac{(r + 0.6) (r + 0.6) - 2 r (r + 0.6) \frac{d}{d r} [r + 0.6]}{{(r + 0.6)}^{4}}$

Step 3.5.2.2

Factor $r + 0.6$ out of $- 2 r (r + 0.6) \frac{d}{d r} [r + 0.6]$ .

$144 \frac{(r + 0.6) (r + 0.6) + (r + 0.6) (- 2 r \frac{d}{d r} [r + 0.6])}{{(r + 0.6)}^{4}}$

Step 3.5.2.3

Factor $r + 0.6$ out of $(r + 0.6) (r + 0.6) + (r + 0.6) (- 2 r \frac{d}{d r} [r + 0.6])$ .

$144 \frac{(r + 0.6) (r + 0.6 - 2 r \frac{d}{d r} [r + 0.6])}{{(r + 0.6)}^{4}}$

Step 3.6

Cancel the common factors.

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Step 3.6.1

Factor $r + 0.6$ out of ${(r + 0.6)}^{4}$ .

$144 \frac{(r + 0.6) (r + 0.6 - 2 r \frac{d}{d r} [r + 0.6])}{(r + 0.6) {(r + 0.6)}^{3}}$

Step 3.6.2

Cancel the common factor.

$144 \frac{(r + 0.6) (r + 0.6 - 2 r \frac{d}{d r} [r + 0.6])}{(r + 0.6) {(r + 0.6)}^{3}}$

Step 3.6.3

Rewrite the expression.

$144 \frac{r + 0.6 - 2 r \frac{d}{d r} [r + 0.6]}{{(r + 0.6)}^{3}}$

Step 3.7

By the Sum Rule, the derivative of $r + 0.6$ with respect to $r$ is $\frac{d}{d r} [r] + \frac{d}{d r} [0.6]$ .

$144 \frac{r + 0.6 - 2 r (\frac{d}{d r} [r] + \frac{d}{d r} [0.6])}{{(r + 0.6)}^{3}}$

Step 3.8

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

$144 \frac{r + 0.6 - 2 r (1 + \frac{d}{d r} [0.6])}{{(r + 0.6)}^{3}}$

Step 3.9

Since $0.6$ is constant with respect to $r$ , the derivative of $0.6$ with respect to $r$ is $0$ .

$144 \frac{r + 0.6 - 2 r (1 + 0)}{{(r + 0.6)}^{3}}$

Step 3.10

Simplify terms.

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Step 3.10.1

Add $1$ and $0$ .

$144 \frac{r + 0.6 - 2 r \cdot 1}{{(r + 0.6)}^{3}}$

Step 3.10.2

Multiply $- 2$ by $1$ .

$144 \frac{r + 0.6 - 2 r}{{(r + 0.6)}^{3}}$

Step 3.10.3

Subtract $2 r$ from $r$ .

$144 \frac{- r + 0.6}{{(r + 0.6)}^{3}}$

Step 3.10.4

Combine $144$ and $\frac{- r + 0.6}{{(r + 0.6)}^{3}}$ .

$\frac{144 (- r + 0.6)}{{(r + 0.6)}^{3}}$

Step 3.11

Simplify.

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Step 3.11.1

Apply the distributive property.

$\frac{144 (- r) + 144 \cdot 0.6}{{(r + 0.6)}^{3}}$

Step 3.11.2

Simplify each term.

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Step 3.11.2.1

Multiply $- 1$ by $144$ .

$\frac{- 144 r + 144 \cdot 0.6}{{(r + 0.6)}^{3}}$

Step 3.11.2.2

Multiply $144$ by $0.6$ .

$\frac{- 144 r + 86.4}{{(r + 0.6)}^{3}}$

Step 3.11.3

Factor $28.8$ out of $- 144 r + 86.4$ .

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Step 3.11.3.1

Factor $28.8$ out of $- 144 r$ .

$\frac{28.8 (- 5 r) + 86.4}{{(r + 0.6)}^{3}}$

Step 3.11.3.2

Factor $28.8$ out of $86.4$ .

$\frac{28.8 (- 5 r) + 28.8 (3)}{{(r + 0.6)}^{3}}$

Step 3.11.3.3

Factor $28.8$ out of $28.8 (- 5 r) + 28.8 (3)$ .

$\frac{28.8 (- 5 r + 3)}{{(r + 0.6)}^{3}}$

Step 3.11.4

Factor $- 1$ out of $- 5 r$ .

$\frac{28.8 (- (5 r) + 3)}{{(r + 0.6)}^{3}}$

Step 3.11.5

Rewrite $3$ as $- 1 (- 3)$ .

$\frac{28.8 (- (5 r) - 1 (- 3))}{{(r + 0.6)}^{3}}$

Step 3.11.6

Factor $- 1$ out of $- (5 r) - 1 (- 3)$ .

$\frac{28.8 (- (5 r - 3))}{{(r + 0.6)}^{3}}$

Step 3.11.7

Rewrite $- (5 r - 3)$ as $- 1 (5 r - 3)$ .

$\frac{28.8 (- 1 (5 r - 3))}{{(r + 0.6)}^{3}}$

Step 3.11.8

Move the negative in front of the fraction.

$- \frac{28.8 (5 r - 3)}{{(r + 0.6)}^{3}}$

Step 4

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

$p' = - \frac{28.8 (5 r - 3)}{{(r + 0.6)}^{3}}$

Step 5

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

$\frac{d p}{d r} = - \frac{28.8 (5 r - 3)}{{(r + 0.6)}^{3}}$