Calculus Examples

$g (r) = r^{2} \ln (2 r + 1)$

Step 1

Differentiate using the Product Rule which states that $\frac{d}{d r} [f (r) g (r)]$ is $f (r) \frac{d}{d r} [g (r)] + g (r) \frac{d}{d r} [f (r)]$ where $f (r) = r^{2}$ and $g (r) = \ln (2 r + 1)$ .

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

Step 2

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

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Step 2.1

To apply the Chain Rule, set $u$ as $2 r + 1$ .

$r^{2} (\frac{d}{d u} [\ln (u)] \frac{d}{d r} [2 r + 1]) + \ln (2 r + 1) \frac{d}{d r} [r^{2}]$

Step 2.2

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

$r^{2} (\frac{1}{u} \frac{d}{d r} [2 r + 1]) + \ln (2 r + 1) \frac{d}{d r} [r^{2}]$

Step 2.3

Replace all occurrences of $u$ with $2 r + 1$ .

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

Step 3

Differentiate.

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Step 3.1

Combine $\frac{1}{2 r + 1}$ and $r^{2}$ .

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

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

$\frac{r^{2}}{2 r + 1} (2 \cdot 1 + \frac{d}{d r} [1]) + \ln (2 r + 1) \frac{d}{d r} [r^{2}]$

Step 3.5

Multiply $2$ by $1$ .

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

Step 3.6

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

$\frac{r^{2}}{2 r + 1} (2 + 0) + \ln (2 r + 1) \frac{d}{d r} [r^{2}]$

Step 3.7

Combine fractions.

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Step 3.7.1

Add $2$ and $0$ .

$\frac{r^{2}}{2 r + 1} \cdot 2 + \ln (2 r + 1) \frac{d}{d r} [r^{2}]$

Step 3.7.2

Combine $\frac{r^{2}}{2 r + 1}$ and $2$ .

$\frac{r^{2} \cdot 2}{2 r + 1} + \ln (2 r + 1) \frac{d}{d r} [r^{2}]$

Step 3.7.3

Move $2$ to the left of $r^{2}$ .

$\frac{2 \cdot r^{2}}{2 r + 1} + \ln (2 r + 1) \frac{d}{d r} [r^{2}]$

Step 3.8

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

$\frac{2 r^{2}}{2 r + 1} + \ln (2 r + 1) (2 r)$

Step 4

To write $\ln (2 r + 1) (2 r)$ as a fraction with a common denominator, multiply by $\frac{2 r + 1}{2 r + 1}$ .

$\frac{2 r^{2}}{2 r + 1} + \frac{\ln (2 r + 1) (2 r) (2 r + 1)}{2 r + 1}$

Step 5

Combine the numerators over the common denominator.

$\frac{2 r^{2} + \ln (2 r + 1) (2 r) (2 r + 1)}{2 r + 1}$

Step 6

Simplify the numerator.

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Step 6.1

Simplify each term.

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Step 6.1.1

Rewrite using the commutative property of multiplication.

$\frac{2 r^{2} + 2 \ln (2 r + 1) r (2 r + 1)}{2 r + 1}$

Step 6.1.2

Simplify $2 \ln (2 r + 1)$ by moving $2$ inside the logarithm.

$\frac{2 r^{2} + \ln ({(2 r + 1)}^{2}) r (2 r + 1)}{2 r + 1}$

Step 6.1.3

Apply the distributive property.

$\frac{2 r^{2} + \ln ({(2 r + 1)}^{2}) r (2 r) + \ln ({(2 r + 1)}^{2}) r \cdot 1}{2 r + 1}$

Step 6.1.4

Rewrite using the commutative property of multiplication.

$\frac{2 r^{2} + 2 \ln ({(2 r + 1)}^{2}) r \cdot r + \ln ({(2 r + 1)}^{2}) r \cdot 1}{2 r + 1}$

Step 6.1.5

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

$\frac{2 r^{2} + 2 \ln ({(2 r + 1)}^{2}) r \cdot r + r \cdot \ln ({(2 r + 1)}^{2})}{2 r + 1}$

Step 6.1.6

Simplify each term.

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Step 6.1.6.1

Multiply $r$ by $r$ by adding the exponents.

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Step 6.1.6.1.1

Move $r$ .

$\frac{2 r^{2} + 2 \ln ({(2 r + 1)}^{2}) (r \cdot r) + r \cdot \ln ({(2 r + 1)}^{2})}{2 r + 1}$

Step 6.1.6.1.2

Multiply $r$ by $r$ .

$\frac{2 r^{2} + 2 \ln ({(2 r + 1)}^{2}) r^{2} + r \cdot \ln ({(2 r + 1)}^{2})}{2 r + 1}$

Step 6.1.6.2

Simplify $2 \ln ({(2 r + 1)}^{2})$ by moving $2$ inside the logarithm.

$\frac{2 r^{2} + \ln ({({(2 r + 1)}^{2})}^{2}) r^{2} + r \cdot \ln ({(2 r + 1)}^{2})}{2 r + 1}$

Step 6.1.6.3

Multiply the exponents in ${({(2 r + 1)}^{2})}^{2}$ .

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Step 6.1.6.3.1

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

$\frac{2 r^{2} + \ln ({(2 r + 1)}^{2 \cdot 2}) r^{2} + r \cdot \ln ({(2 r + 1)}^{2})}{2 r + 1}$

Step 6.1.6.3.2

Multiply $2$ by $2$ .

$\frac{2 r^{2} + \ln ({(2 r + 1)}^{4}) r^{2} + r \cdot \ln ({(2 r + 1)}^{2})}{2 r + 1}$

$\frac{2 r^{2} + \ln ({(2 r + 1)}^{4}) r^{2} + r \ln ({(2 r + 1)}^{2})}{2 r + 1}$

Step 6.2

Reorder factors in $2 r^{2} + \ln ({(2 r + 1)}^{4}) r^{2} + r \ln ({(2 r + 1)}^{2})$ .

$\frac{2 r^{2} + r^{2} \ln ({(2 r + 1)}^{4}) + r \ln ({(2 r + 1)}^{2})}{2 r + 1}$