Calculus Examples

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

Step 1

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 $f (x) = {(4 x - 3)}^{2}$ and $g (x) = {(4 - x^{5})}^{4}$ .

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

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{4}$ and $g (x) = 4 - x^{5}$ .

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Step 2.1

To apply the Chain Rule, set $u$ as $4 - x^{5}$ .

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

Step 2.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 4$ .

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

Step 2.3

Replace all occurrences of $u$ with $4 - x^{5}$ .

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

Step 3

Differentiate.

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Step 3.1

By the Sum Rule, the derivative of $4 - x^{5}$ with respect to $x$ is $\frac{d}{d x} [4] + \frac{d}{d x} [- x^{5}]$ .

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

Step 3.2

Since $4$ is constant with respect to $x$ , the derivative of $4$ with respect to $x$ is $0$ .

${(4 x - 3)}^{2} (4 {(4 - x^{5})}^{3} (0 + \frac{d}{d x} [- x^{5}])) + {(4 - x^{5})}^{4} \frac{d}{d x} [{(4 x - 3)}^{2}]$

Step 3.3

Add $0$ and $\frac{d}{d x} [- x^{5}]$ .

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

Step 3.4

Since $- 1$ is constant with respect to $x$ , the derivative of $- x^{5}$ with respect to $x$ is $- \frac{d}{d x} [x^{5}]$ .

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

Step 3.5

Multiply $- 1$ by $4$ .

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

Step 3.6

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 5$ .

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

Step 3.7

Simplify the expression.

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Step 3.7.1

Multiply $5$ by $- 4$ .

${(4 x - 3)}^{2} (- 20 {(4 - x^{5})}^{3} x^{4}) + {(4 - x^{5})}^{4} \frac{d}{d x} [{(4 x - 3)}^{2}]$

Step 3.7.2

Reorder the factors of $- 20 {(4 - x^{5})}^{3} x^{4}$ .

${(4 x - 3)}^{2} (- 20 x^{4} {(4 - x^{5})}^{3}) + {(4 - x^{5})}^{4} \frac{d}{d x} [{(4 x - 3)}^{2}]$

Step 3.7.3

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

${(4 x - 3)}^{2} (- 20 x^{4} {(4 - x^{5})}^{3}) + {(4 - x^{5})}^{4} \frac{d}{d x} [(4 x - 3) (4 x - 3)]$

Step 4

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

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Step 4.1

Apply the distributive property.

${(4 x - 3)}^{2} (- 20 x^{4} {(4 - x^{5})}^{3}) + {(4 - x^{5})}^{4} \frac{d}{d x} [4 x (4 x - 3) - 3 (4 x - 3)]$

Step 4.2

Apply the distributive property.

${(4 x - 3)}^{2} (- 20 x^{4} {(4 - x^{5})}^{3}) + {(4 - x^{5})}^{4} \frac{d}{d x} [4 x (4 x) + 4 x \cdot - 3 - 3 (4 x - 3)]$

Step 4.3

Apply the distributive property.

${(4 x - 3)}^{2} (- 20 x^{4} {(4 - x^{5})}^{3}) + {(4 - x^{5})}^{4} \frac{d}{d x} [4 x (4 x) + 4 x \cdot - 3 - 3 (4 x) - 3 \cdot - 3]$

Step 5

Simplify and combine like terms.

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Step 5.1

Simplify each term.

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Step 5.1.1

Rewrite using the commutative property of multiplication.

${(4 x - 3)}^{2} (- 20 x^{4} {(4 - x^{5})}^{3}) + {(4 - x^{5})}^{4} \frac{d}{d x} [4 \cdot 4 x \cdot x + 4 x \cdot - 3 - 3 (4 x) - 3 \cdot - 3]$

Step 5.1.2

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

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Step 5.1.2.1

Move $x$ .

${(4 x - 3)}^{2} (- 20 x^{4} {(4 - x^{5})}^{3}) + {(4 - x^{5})}^{4} \frac{d}{d x} [4 \cdot 4 (x \cdot x) + 4 x \cdot - 3 - 3 (4 x) - 3 \cdot - 3]$

Step 5.1.2.2

Multiply $x$ by $x$ .

${(4 x - 3)}^{2} (- 20 x^{4} {(4 - x^{5})}^{3}) + {(4 - x^{5})}^{4} \frac{d}{d x} [4 \cdot 4 x^{2} + 4 x \cdot - 3 - 3 (4 x) - 3 \cdot - 3]$

Step 5.1.3

Multiply $4$ by $4$ .

${(4 x - 3)}^{2} (- 20 x^{4} {(4 - x^{5})}^{3}) + {(4 - x^{5})}^{4} \frac{d}{d x} [16 x^{2} + 4 x \cdot - 3 - 3 (4 x) - 3 \cdot - 3]$

Step 5.1.4

Multiply $- 3$ by $4$ .

${(4 x - 3)}^{2} (- 20 x^{4} {(4 - x^{5})}^{3}) + {(4 - x^{5})}^{4} \frac{d}{d x} [16 x^{2} - 12 x - 3 (4 x) - 3 \cdot - 3]$

Step 5.1.5

Multiply $4$ by $- 3$ .

${(4 x - 3)}^{2} (- 20 x^{4} {(4 - x^{5})}^{3}) + {(4 - x^{5})}^{4} \frac{d}{d x} [16 x^{2} - 12 x - 12 x - 3 \cdot - 3]$

Step 5.1.6

Multiply $- 3$ by $- 3$ .

${(4 x - 3)}^{2} (- 20 x^{4} {(4 - x^{5})}^{3}) + {(4 - x^{5})}^{4} \frac{d}{d x} [16 x^{2} - 12 x - 12 x + 9]$

Step 5.2

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

${(4 x - 3)}^{2} (- 20 x^{4} {(4 - x^{5})}^{3}) + {(4 - x^{5})}^{4} \frac{d}{d x} [16 x^{2} - 24 x + 9]$

Step 6

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

${(4 x - 3)}^{2} (- 20 x^{4} {(4 - x^{5})}^{3}) + {(4 - x^{5})}^{4} (\frac{d}{d x} [16 x^{2}] + \frac{d}{d x} [- 24 x] + \frac{d}{d x} [9])$

Step 7

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

${(4 x - 3)}^{2} (- 20 x^{4} {(4 - x^{5})}^{3}) + {(4 - x^{5})}^{4} (16 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 24 x] + \frac{d}{d x} [9])$

Step 8

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

${(4 x - 3)}^{2} (- 20 x^{4} {(4 - x^{5})}^{3}) + {(4 - x^{5})}^{4} (16 (2 x) + \frac{d}{d x} [- 24 x] + \frac{d}{d x} [9])$

Step 9

Multiply $2$ by $16$ .

${(4 x - 3)}^{2} (- 20 x^{4} {(4 - x^{5})}^{3}) + {(4 - x^{5})}^{4} (32 x + \frac{d}{d x} [- 24 x] + \frac{d}{d x} [9])$

Step 10

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

${(4 x - 3)}^{2} (- 20 x^{4} {(4 - x^{5})}^{3}) + {(4 - x^{5})}^{4} (32 x - 24 \frac{d}{d x} [x] + \frac{d}{d x} [9])$

Step 11

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

${(4 x - 3)}^{2} (- 20 x^{4} {(4 - x^{5})}^{3}) + {(4 - x^{5})}^{4} (32 x - 24 \cdot 1 + \frac{d}{d x} [9])$

Step 12

Multiply $- 24$ by $1$ .

${(4 x - 3)}^{2} (- 20 x^{4} {(4 - x^{5})}^{3}) + {(4 - x^{5})}^{4} (32 x - 24 + \frac{d}{d x} [9])$

Step 13

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

${(4 x - 3)}^{2} (- 20 x^{4} {(4 - x^{5})}^{3}) + {(4 - x^{5})}^{4} (32 x - 24 + 0)$

Step 14

Simplify with factoring out.

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Step 14.1

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

${(4 x - 3)}^{2} (- 20 x^{4} {(4 - x^{5})}^{3}) + {(4 - x^{5})}^{4} (32 x - 24)$

Step 14.2

Factor ${(4 - x^{5})}^{3}$ out of ${(4 x - 3)}^{2} (- 20 x^{4} {(4 - x^{5})}^{3}) + {(4 - x^{5})}^{4} (32 x - 24)$ .

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Step 14.2.1

Factor ${(4 - x^{5})}^{3}$ out of ${(4 x - 3)}^{2} (- 20 x^{4} {(4 - x^{5})}^{3})$ .

${(4 - x^{5})}^{3} ({(4 x - 3)}^{2} (- 20 x^{4})) + {(4 - x^{5})}^{4} (32 x - 24)$

Step 14.2.2

Factor ${(4 - x^{5})}^{3}$ out of ${(4 - x^{5})}^{4} (32 x - 24)$ .

${(4 - x^{5})}^{3} ({(4 x - 3)}^{2} (- 20 x^{4})) + {(4 - x^{5})}^{3} ((4 - x^{5}) (32 x - 24))$

Step 14.2.3

Factor ${(4 - x^{5})}^{3}$ out of ${(4 - x^{5})}^{3} ({(4 x - 3)}^{2} (- 20 x^{4})) + {(4 - x^{5})}^{3} ((4 - x^{5}) (32 x - 24))$ .

${(4 - x^{5})}^{3} ({(4 x - 3)}^{2} (- 20 x^{4}) + (4 - x^{5}) (32 x - 24))$