Calculus Examples

$y = (16 x + 5) {(4 x - 3)}^{2}$

Step 1

Rewrite ${(4 x - 3)}^{2}$ as $(4 x - 3) (4 x - 3)$ .

$\frac{d}{d x} [(16 x + 5) ((4 x - 3) (4 x - 3))]$

Step 2

Expand $(4 x - 3) (4 x - 3)$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

$\frac{d}{d x} [(16 x + 5) (4 x (4 x - 3) - 3 (4 x - 3))]$

Step 2.2

Apply the distributive property.

$\frac{d}{d x} [(16 x + 5) (4 x (4 x) + 4 x \cdot - 3 - 3 (4 x - 3))]$

Step 2.3

Apply the distributive property.

$\frac{d}{d x} [(16 x + 5) (4 x (4 x) + 4 x \cdot - 3 - 3 (4 x) - 3 \cdot - 3)]$

Step 3

Simplify and combine like terms.

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Step 3.1

Simplify each term.

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Step 3.1.1

Rewrite using the commutative property of multiplication.

$\frac{d}{d x} [(16 x + 5) (4 \cdot 4 x \cdot x + 4 x \cdot - 3 - 3 (4 x) - 3 \cdot - 3)]$

Step 3.1.2

Multiply $x$ by $x$ by adding the exponents.

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Step 3.1.2.1

Move $x$ .

$\frac{d}{d x} [(16 x + 5) (4 \cdot 4 (x \cdot x) + 4 x \cdot - 3 - 3 (4 x) - 3 \cdot - 3)]$

Step 3.1.2.2

Multiply $x$ by $x$ .

$\frac{d}{d x} [(16 x + 5) (4 \cdot 4 x^{2} + 4 x \cdot - 3 - 3 (4 x) - 3 \cdot - 3)]$

Step 3.1.3

Multiply $4$ by $4$ .

$\frac{d}{d x} [(16 x + 5) (16 x^{2} + 4 x \cdot - 3 - 3 (4 x) - 3 \cdot - 3)]$

Step 3.1.4

Multiply $- 3$ by $4$ .

$\frac{d}{d x} [(16 x + 5) (16 x^{2} - 12 x - 3 (4 x) - 3 \cdot - 3)]$

Step 3.1.5

Multiply $4$ by $- 3$ .

$\frac{d}{d x} [(16 x + 5) (16 x^{2} - 12 x - 12 x - 3 \cdot - 3)]$

Step 3.1.6

Multiply $- 3$ by $- 3$ .

$\frac{d}{d x} [(16 x + 5) (16 x^{2} - 12 x - 12 x + 9)]$

Step 3.2

Subtract $12 x$ from $- 12 x$ .

$\frac{d}{d x} [(16 x + 5) (16 x^{2} - 24 x + 9)]$

Step 4

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = 16 x + 5$ and $g (x) = 16 x^{2} - 24 x + 9$ .

$(16 x + 5) \frac{d}{d x} [16 x^{2} - 24 x + 9] + (16 x^{2} - 24 x + 9) \frac{d}{d x} [16 x + 5]$

Step 5

Differentiate.

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Step 5.1

By the Sum Rule, the derivative of $16 x^{2} - 24 x + 9$ with respect to $x$ is $\frac{d}{d x} [16 x^{2}] + \frac{d}{d x} [- 24 x] + \frac{d}{d x} [9]$ .

$(16 x + 5) (\frac{d}{d x} [16 x^{2}] + \frac{d}{d x} [- 24 x] + \frac{d}{d x} [9]) + (16 x^{2} - 24 x + 9) \frac{d}{d x} [16 x + 5]$

Step 5.2

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

$(16 x + 5) (16 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 24 x] + \frac{d}{d x} [9]) + (16 x^{2} - 24 x + 9) \frac{d}{d x} [16 x + 5]$

Step 5.3

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

$(16 x + 5) (16 (2 x) + \frac{d}{d x} [- 24 x] + \frac{d}{d x} [9]) + (16 x^{2} - 24 x + 9) \frac{d}{d x} [16 x + 5]$

Step 5.4

Multiply $2$ by $16$ .

$(16 x + 5) (32 x + \frac{d}{d x} [- 24 x] + \frac{d}{d x} [9]) + (16 x^{2} - 24 x + 9) \frac{d}{d x} [16 x + 5]$

Step 5.5

Since $- 24$ is constant with respect to $x$ , the derivative of $- 24 x$ with respect to $x$ is $- 24 \frac{d}{d x} [x]$ .

$(16 x + 5) (32 x - 24 \frac{d}{d x} [x] + \frac{d}{d x} [9]) + (16 x^{2} - 24 x + 9) \frac{d}{d x} [16 x + 5]$

Step 5.6

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

$(16 x + 5) (32 x - 24 \cdot 1 + \frac{d}{d x} [9]) + (16 x^{2} - 24 x + 9) \frac{d}{d x} [16 x + 5]$

Step 5.7

Multiply $- 24$ by $1$ .

$(16 x + 5) (32 x - 24 + \frac{d}{d x} [9]) + (16 x^{2} - 24 x + 9) \frac{d}{d x} [16 x + 5]$

Step 5.8

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

$(16 x + 5) (32 x - 24 + 0) + (16 x^{2} - 24 x + 9) \frac{d}{d x} [16 x + 5]$

Step 5.9

Add $32 x - 24$ and $0$ .

$(16 x + 5) (32 x - 24) + (16 x^{2} - 24 x + 9) \frac{d}{d x} [16 x + 5]$

Step 5.10

By the Sum Rule, the derivative of $16 x + 5$ with respect to $x$ is $\frac{d}{d x} [16 x] + \frac{d}{d x} [5]$ .

$(16 x + 5) (32 x - 24) + (16 x^{2} - 24 x + 9) (\frac{d}{d x} [16 x] + \frac{d}{d x} [5])$

Step 5.11

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

$(16 x + 5) (32 x - 24) + (16 x^{2} - 24 x + 9) (16 \frac{d}{d x} [x] + \frac{d}{d x} [5])$

Step 5.12

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

$(16 x + 5) (32 x - 24) + (16 x^{2} - 24 x + 9) (16 \cdot 1 + \frac{d}{d x} [5])$

Step 5.13

Multiply $16$ by $1$ .

$(16 x + 5) (32 x - 24) + (16 x^{2} - 24 x + 9) (16 + \frac{d}{d x} [5])$

Step 5.14

Since $5$ is constant with respect to $x$ , the derivative of $5$ with respect to $x$ is $0$ .

$(16 x + 5) (32 x - 24) + (16 x^{2} - 24 x + 9) (16 + 0)$

Step 5.15

Simplify the expression.

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Step 5.15.1

Add $16$ and $0$ .

$(16 x + 5) (32 x - 24) + (16 x^{2} - 24 x + 9) \cdot 16$

Step 5.15.2

Move $16$ to the left of $16 x^{2} - 24 x + 9$ .

$(16 x + 5) (32 x - 24) + 16 \cdot (16 x^{2} - 24 x + 9)$

Step 6

Simplify.

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Step 6.1

Apply the distributive property.

$(16 x + 5) (32 x - 24) + 16 (16 x^{2}) + 16 (- 24 x) + 16 \cdot 9$

Step 6.2

Apply the distributive property.

$(16 x + 5) (32 x) + (16 x + 5) \cdot - 24 + 16 (16 x^{2}) + 16 (- 24 x) + 16 \cdot 9$

Step 6.3

Apply the distributive property.

$16 x (32 x) + 5 (32 x) + (16 x + 5) \cdot - 24 + 16 (16 x^{2}) + 16 (- 24 x) + 16 \cdot 9$

Step 6.4

Apply the distributive property.

$16 x (32 x) + 5 (32 x) + 16 x \cdot - 24 + 5 \cdot - 24 + 16 (16 x^{2}) + 16 (- 24 x) + 16 \cdot 9$

Step 6.5

Combine terms.

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Step 6.5.1

Multiply $32$ by $16$ .

$512 x \cdot x + 5 (32 x) + 16 x \cdot - 24 + 5 \cdot - 24 + 16 (16 x^{2}) + 16 (- 24 x) + 16 \cdot 9$

Step 6.5.2

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

$512 (x^{1} x) + 5 (32 x) + 16 x \cdot - 24 + 5 \cdot - 24 + 16 (16 x^{2}) + 16 (- 24 x) + 16 \cdot 9$

Step 6.5.3

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

$512 (x^{1} x^{1}) + 5 (32 x) + 16 x \cdot - 24 + 5 \cdot - 24 + 16 (16 x^{2}) + 16 (- 24 x) + 16 \cdot 9$

Step 6.5.4

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

$512 x^{1 + 1} + 5 (32 x) + 16 x \cdot - 24 + 5 \cdot - 24 + 16 (16 x^{2}) + 16 (- 24 x) + 16 \cdot 9$

Step 6.5.5

Add $1$ and $1$ .

$512 x^{2} + 5 (32 x) + 16 x \cdot - 24 + 5 \cdot - 24 + 16 (16 x^{2}) + 16 (- 24 x) + 16 \cdot 9$

Step 6.5.6

Multiply $32$ by $5$ .

$512 x^{2} + 160 x + 16 x \cdot - 24 + 5 \cdot - 24 + 16 (16 x^{2}) + 16 (- 24 x) + 16 \cdot 9$

Step 6.5.7

Multiply $- 24$ by $16$ .

$512 x^{2} + 160 x - 384 x + 5 \cdot - 24 + 16 (16 x^{2}) + 16 (- 24 x) + 16 \cdot 9$

Step 6.5.8

Multiply $5$ by $- 24$ .

$512 x^{2} + 160 x - 384 x - 120 + 16 (16 x^{2}) + 16 (- 24 x) + 16 \cdot 9$

Step 6.5.9

Subtract $384 x$ from $160 x$ .

$512 x^{2} - 224 x - 120 + 16 (16 x^{2}) + 16 (- 24 x) + 16 \cdot 9$

Step 6.5.10

Multiply $16$ by $16$ .

$512 x^{2} - 224 x - 120 + 256 x^{2} + 16 (- 24 x) + 16 \cdot 9$

Step 6.5.11

Multiply $- 24$ by $16$ .

$512 x^{2} - 224 x - 120 + 256 x^{2} - 384 x + 16 \cdot 9$

Step 6.5.12

Multiply $16$ by $9$ .

$512 x^{2} - 224 x - 120 + 256 x^{2} - 384 x + 144$

Step 6.5.13

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

$768 x^{2} - 224 x - 120 - 384 x + 144$

Step 6.5.14

Subtract $384 x$ from $- 224 x$ .

$768 x^{2} - 608 x - 120 + 144$

Step 6.5.15

Add $- 120$ and $144$ .

$768 x^{2} - 608 x + 24$