Calculus Examples

$f (r) = r {(4 r + 9)}^{3}$

Step 1

Find the first derivative.

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Step 1.1

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 $f (r) = r$ and $g (r) = {(4 r + 9)}^{3}$ .

$r \frac{d}{d r} [{(4 r + 9)}^{3}] + {(4 r + 9)}^{3} \frac{d}{d r} [r]$

Step 1.2

Differentiate using the chain rule, which states that $\frac{d}{d r} [f (g (r))]$ is $f' (g (r)) g' (r)$ where $f (r) = r^{3}$ and $g (r) = 4 r + 9$ .

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Step 1.2.1

To apply the Chain Rule, set $u$ as $4 r + 9$ .

$r (\frac{d}{d u} [u^{3}] \frac{d}{d r} [4 r + 9]) + {(4 r + 9)}^{3} \frac{d}{d r} [r]$

Step 1.2.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 3$ .

$r (3 u^{2} \frac{d}{d r} [4 r + 9]) + {(4 r + 9)}^{3} \frac{d}{d r} [r]$

Step 1.2.3

Replace all occurrences of $u$ with $4 r + 9$ .

$r (3 {(4 r + 9)}^{2} \frac{d}{d r} [4 r + 9]) + {(4 r + 9)}^{3} \frac{d}{d r} [r]$

Step 1.3

Differentiate.

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Step 1.3.1

By the Sum Rule, the derivative of $4 r + 9$ with respect to $r$ is $\frac{d}{d r} [4 r] + \frac{d}{d r} [9]$ .

$r (3 {(4 r + 9)}^{2} (\frac{d}{d r} [4 r] + \frac{d}{d r} [9])) + {(4 r + 9)}^{3} \frac{d}{d r} [r]$

Step 1.3.2

Since $4$ is constant with respect to $r$ , the derivative of $4 r$ with respect to $r$ is $4 \frac{d}{d r} [r]$ .

$r (3 {(4 r + 9)}^{2} (4 \frac{d}{d r} [r] + \frac{d}{d r} [9])) + {(4 r + 9)}^{3} \frac{d}{d r} [r]$

Step 1.3.3

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

$r (3 {(4 r + 9)}^{2} (4 \cdot 1 + \frac{d}{d r} [9])) + {(4 r + 9)}^{3} \frac{d}{d r} [r]$

Step 1.3.4

Multiply $4$ by $1$ .

$r (3 {(4 r + 9)}^{2} (4 + \frac{d}{d r} [9])) + {(4 r + 9)}^{3} \frac{d}{d r} [r]$

Step 1.3.5

Since $9$ is constant with respect to $r$ , the derivative of $9$ with respect to $r$ is $0$ .

$r (3 {(4 r + 9)}^{2} (4 + 0)) + {(4 r + 9)}^{3} \frac{d}{d r} [r]$

Step 1.3.6

Simplify the expression.

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Step 1.3.6.1

Add $4$ and $0$ .

$r (3 {(4 r + 9)}^{2} \cdot 4) + {(4 r + 9)}^{3} \frac{d}{d r} [r]$

Step 1.3.6.2

Multiply $4$ by $3$ .

$r (12 {(4 r + 9)}^{2}) + {(4 r + 9)}^{3} \frac{d}{d r} [r]$

Step 1.3.7

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

$r (12 {(4 r + 9)}^{2}) + {(4 r + 9)}^{3} \cdot 1$

Step 1.3.8

Multiply ${(4 r + 9)}^{3}$ by $1$ .

$r (12 {(4 r + 9)}^{2}) + {(4 r + 9)}^{3}$

Step 1.4

Simplify.

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Step 1.4.1

Factor ${(4 r + 9)}^{2}$ out of $r (12 {(4 r + 9)}^{2}) + {(4 r + 9)}^{3}$ .

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Step 1.4.1.1

Factor ${(4 r + 9)}^{2}$ out of $r (12 {(4 r + 9)}^{2})$ .

${(4 r + 9)}^{2} (r \cdot (12)) + {(4 r + 9)}^{3}$

Step 1.4.1.2

Factor ${(4 r + 9)}^{2}$ out of ${(4 r + 9)}^{3}$ .

${(4 r + 9)}^{2} (r \cdot (12)) + {(4 r + 9)}^{2} (4 r + 9)$

Step 1.4.1.3

Factor ${(4 r + 9)}^{2}$ out of ${(4 r + 9)}^{2} (r \cdot (12)) + {(4 r + 9)}^{2} (4 r + 9)$ .

${(4 r + 9)}^{2} (r \cdot (12) + 4 r + 9)$

Step 1.4.2

Combine terms.

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Step 1.4.2.1

Move $12$ to the left of $r$ .

${(4 r + 9)}^{2} (12 \cdot r + 4 r + 9)$

Step 1.4.2.2

Add $12 r$ and $4 r$ .

$f' (r) = {(4 r + 9)}^{2} (16 r + 9)$

Step 2

Find the second derivative.

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Step 2.1

Rewrite ${(4 r + 9)}^{2}$ as $(4 r + 9) (4 r + 9)$ .

$\frac{d}{d r} [(4 r + 9) (4 r + 9) (16 r + 9)]$

Step 2.2

Expand $(4 r + 9) (4 r + 9)$ using the FOIL Method.

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Step 2.2.1

Apply the distributive property.

$\frac{d}{d r} [(4 r (4 r + 9) + 9 (4 r + 9)) (16 r + 9)]$

Step 2.2.2

Apply the distributive property.

$\frac{d}{d r} [(4 r (4 r) + 4 r \cdot 9 + 9 (4 r + 9)) (16 r + 9)]$

Step 2.2.3

Apply the distributive property.

$\frac{d}{d r} [(4 r (4 r) + 4 r \cdot 9 + 9 (4 r) + 9 \cdot 9) (16 r + 9)]$

Step 2.3

Simplify and combine like terms.

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Step 2.3.1

Simplify each term.

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Step 2.3.1.1

Rewrite using the commutative property of multiplication.

$\frac{d}{d r} [(4 \cdot 4 r \cdot r + 4 r \cdot 9 + 9 (4 r) + 9 \cdot 9) (16 r + 9)]$

Step 2.3.1.2

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

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Step 2.3.1.2.1

Move $r$ .

$\frac{d}{d r} [(4 \cdot 4 (r \cdot r) + 4 r \cdot 9 + 9 (4 r) + 9 \cdot 9) (16 r + 9)]$

Step 2.3.1.2.2

Multiply $r$ by $r$ .

$\frac{d}{d r} [(4 \cdot 4 r^{2} + 4 r \cdot 9 + 9 (4 r) + 9 \cdot 9) (16 r + 9)]$

Step 2.3.1.3

Multiply $4$ by $4$ .

$\frac{d}{d r} [(16 r^{2} + 4 r \cdot 9 + 9 (4 r) + 9 \cdot 9) (16 r + 9)]$

Step 2.3.1.4

Multiply $9$ by $4$ .

$\frac{d}{d r} [(16 r^{2} + 36 r + 9 (4 r) + 9 \cdot 9) (16 r + 9)]$

Step 2.3.1.5

Multiply $4$ by $9$ .

$\frac{d}{d r} [(16 r^{2} + 36 r + 36 r + 9 \cdot 9) (16 r + 9)]$

Step 2.3.1.6

Multiply $9$ by $9$ .

$\frac{d}{d r} [(16 r^{2} + 36 r + 36 r + 81) (16 r + 9)]$

Step 2.3.2

Add $36 r$ and $36 r$ .

$\frac{d}{d r} [(16 r^{2} + 72 r + 81) (16 r + 9)]$

Step 2.4

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 $f (r) = 16 r^{2} + 72 r + 81$ and $g (r) = 16 r + 9$ .

$(16 r^{2} + 72 r + 81) \frac{d}{d r} [16 r + 9] + (16 r + 9) \frac{d}{d r} [16 r^{2} + 72 r + 81]$

Step 2.5

Differentiate.

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Step 2.5.1

By the Sum Rule, the derivative of $16 r + 9$ with respect to $r$ is $\frac{d}{d r} [16 r] + \frac{d}{d r} [9]$ .

$(16 r^{2} + 72 r + 81) (\frac{d}{d r} [16 r] + \frac{d}{d r} [9]) + (16 r + 9) \frac{d}{d r} [16 r^{2} + 72 r + 81]$

Step 2.5.2

Since $16$ is constant with respect to $r$ , the derivative of $16 r$ with respect to $r$ is $16 \frac{d}{d r} [r]$ .

$(16 r^{2} + 72 r + 81) (16 \frac{d}{d r} [r] + \frac{d}{d r} [9]) + (16 r + 9) \frac{d}{d r} [16 r^{2} + 72 r + 81]$

Step 2.5.3

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

$(16 r^{2} + 72 r + 81) (16 \cdot 1 + \frac{d}{d r} [9]) + (16 r + 9) \frac{d}{d r} [16 r^{2} + 72 r + 81]$

Step 2.5.4

Multiply $16$ by $1$ .

$(16 r^{2} + 72 r + 81) (16 + \frac{d}{d r} [9]) + (16 r + 9) \frac{d}{d r} [16 r^{2} + 72 r + 81]$

Step 2.5.5

Since $9$ is constant with respect to $r$ , the derivative of $9$ with respect to $r$ is $0$ .

$(16 r^{2} + 72 r + 81) (16 + 0) + (16 r + 9) \frac{d}{d r} [16 r^{2} + 72 r + 81]$

Step 2.5.6

Simplify the expression.

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Step 2.5.6.1

Add $16$ and $0$ .

$(16 r^{2} + 72 r + 81) \cdot 16 + (16 r + 9) \frac{d}{d r} [16 r^{2} + 72 r + 81]$

Step 2.5.6.2

Move $16$ to the left of $16 r^{2} + 72 r + 81$ .

$16 \cdot (16 r^{2} + 72 r + 81) + (16 r + 9) \frac{d}{d r} [16 r^{2} + 72 r + 81]$

Step 2.5.7

By the Sum Rule, the derivative of $16 r^{2} + 72 r + 81$ with respect to $r$ is $\frac{d}{d r} [16 r^{2}] + \frac{d}{d r} [72 r] + \frac{d}{d r} [81]$ .

$16 (16 r^{2} + 72 r + 81) + (16 r + 9) (\frac{d}{d r} [16 r^{2}] + \frac{d}{d r} [72 r] + \frac{d}{d r} [81])$

Step 2.5.8

Since $16$ is constant with respect to $r$ , the derivative of $16 r^{2}$ with respect to $r$ is $16 \frac{d}{d r} [r^{2}]$ .

$16 (16 r^{2} + 72 r + 81) + (16 r + 9) (16 \frac{d}{d r} [r^{2}] + \frac{d}{d r} [72 r] + \frac{d}{d r} [81])$

Step 2.5.9

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

$16 (16 r^{2} + 72 r + 81) + (16 r + 9) (16 (2 r) + \frac{d}{d r} [72 r] + \frac{d}{d r} [81])$

Step 2.5.10

Multiply $2$ by $16$ .

$16 (16 r^{2} + 72 r + 81) + (16 r + 9) (32 r + \frac{d}{d r} [72 r] + \frac{d}{d r} [81])$

Step 2.5.11

Since $72$ is constant with respect to $r$ , the derivative of $72 r$ with respect to $r$ is $72 \frac{d}{d r} [r]$ .

$16 (16 r^{2} + 72 r + 81) + (16 r + 9) (32 r + 72 \frac{d}{d r} [r] + \frac{d}{d r} [81])$

Step 2.5.12

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

$16 (16 r^{2} + 72 r + 81) + (16 r + 9) (32 r + 72 \cdot 1 + \frac{d}{d r} [81])$

Step 2.5.13

Multiply $72$ by $1$ .

$16 (16 r^{2} + 72 r + 81) + (16 r + 9) (32 r + 72 + \frac{d}{d r} [81])$

Step 2.5.14

Since $81$ is constant with respect to $r$ , the derivative of $81$ with respect to $r$ is $0$ .

$16 (16 r^{2} + 72 r + 81) + (16 r + 9) (32 r + 72 + 0)$

Step 2.5.15

Add $32 r + 72$ and $0$ .

$16 (16 r^{2} + 72 r + 81) + (16 r + 9) (32 r + 72)$

Step 2.6

Simplify.

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Step 2.6.1

Apply the distributive property.

$16 (16 r^{2}) + 16 (72 r) + 16 \cdot 81 + (16 r + 9) (32 r + 72)$

Step 2.6.2

Apply the distributive property.

$16 (16 r^{2}) + 16 (72 r) + 16 \cdot 81 + (16 r + 9) (32 r) + (16 r + 9) \cdot 72$

Step 2.6.3

Apply the distributive property.

$16 (16 r^{2}) + 16 (72 r) + 16 \cdot 81 + 16 r (32 r) + 9 (32 r) + (16 r + 9) \cdot 72$

Step 2.6.4

Apply the distributive property.

$16 (16 r^{2}) + 16 (72 r) + 16 \cdot 81 + 16 r (32 r) + 9 (32 r) + 16 r \cdot 72 + 9 \cdot 72$

Step 2.6.5

Combine terms.

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Step 2.6.5.1

Multiply $16$ by $16$ .

$256 r^{2} + 16 (72 r) + 16 \cdot 81 + 16 r (32 r) + 9 (32 r) + 16 r \cdot 72 + 9 \cdot 72$

Step 2.6.5.2

Multiply $72$ by $16$ .

$256 r^{2} + 1152 r + 16 \cdot 81 + 16 r (32 r) + 9 (32 r) + 16 r \cdot 72 + 9 \cdot 72$

Step 2.6.5.3

Multiply $16$ by $81$ .

$256 r^{2} + 1152 r + 1296 + 16 r (32 r) + 9 (32 r) + 16 r \cdot 72 + 9 \cdot 72$

Step 2.6.5.4

Multiply $32$ by $16$ .

$256 r^{2} + 1152 r + 1296 + 512 r \cdot r + 9 (32 r) + 16 r \cdot 72 + 9 \cdot 72$

Step 2.6.5.5

Raise $r$ to the power of $1$ .

$256 r^{2} + 1152 r + 1296 + 512 (r^{1} r) + 9 (32 r) + 16 r \cdot 72 + 9 \cdot 72$

Step 2.6.5.6

Raise $r$ to the power of $1$ .

$256 r^{2} + 1152 r + 1296 + 512 (r^{1} r^{1}) + 9 (32 r) + 16 r \cdot 72 + 9 \cdot 72$

Step 2.6.5.7

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$256 r^{2} + 1152 r + 1296 + 512 r^{1 + 1} + 9 (32 r) + 16 r \cdot 72 + 9 \cdot 72$

Step 2.6.5.8

Add $1$ and $1$ .

$256 r^{2} + 1152 r + 1296 + 512 r^{2} + 9 (32 r) + 16 r \cdot 72 + 9 \cdot 72$

Step 2.6.5.9

Multiply $32$ by $9$ .

$256 r^{2} + 1152 r + 1296 + 512 r^{2} + 288 r + 16 r \cdot 72 + 9 \cdot 72$

Step 2.6.5.10

Multiply $72$ by $16$ .

$256 r^{2} + 1152 r + 1296 + 512 r^{2} + 288 r + 1152 r + 9 \cdot 72$

Step 2.6.5.11

Multiply $9$ by $72$ .

$256 r^{2} + 1152 r + 1296 + 512 r^{2} + 288 r + 1152 r + 648$

Step 2.6.5.12

Add $288 r$ and $1152 r$ .

$256 r^{2} + 1152 r + 1296 + 512 r^{2} + 1440 r + 648$

Step 2.6.5.13

Add $256 r^{2}$ and $512 r^{2}$ .

$768 r^{2} + 1152 r + 1296 + 1440 r + 648$

Step 2.6.5.14

Add $1152 r$ and $1440 r$ .

$768 r^{2} + 2592 r + 1296 + 648$

Step 2.6.5.15

Add $1296$ and $648$ .

$f'' (r) = 768 r^{2} + 2592 r + 1944$

Step 3

The second derivative of $f (r)$ with respect to $r$ is $768 r^{2} + 2592 r + 1944$ .

$768 r^{2} + 2592 r + 1944$