Calculus Examples

$y = {(6 x - 5)}^{2} {(3 - x^{5})}^{2}$

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 $f (x) = {(6 x - 5)}^{2}$ and $g (x) = {(3 - x^{5})}^{2}$ .

${(6 x - 5)}^{2} \frac{d}{d x} [{(3 - x^{5})}^{2}] + {(3 - x^{5})}^{2} \frac{d}{d x} [{(6 x - 5)}^{2}]$

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{2}$ and $g (x) = 3 - x^{5}$ .

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Step 2.1

To apply the Chain Rule, set $u$ as $3 - x^{5}$ .

${(6 x - 5)}^{2} (\frac{d}{d u} [u^{2}] \frac{d}{d x} [3 - x^{5}]) + {(3 - x^{5})}^{2} \frac{d}{d x} [{(6 x - 5)}^{2}]$

Step 2.2

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

${(6 x - 5)}^{2} (2 u \frac{d}{d x} [3 - x^{5}]) + {(3 - x^{5})}^{2} \frac{d}{d x} [{(6 x - 5)}^{2}]$

Step 2.3

Replace all occurrences of $u$ with $3 - x^{5}$ .

${(6 x - 5)}^{2} (2 (3 - x^{5}) \frac{d}{d x} [3 - x^{5}]) + {(3 - x^{5})}^{2} \frac{d}{d x} [{(6 x - 5)}^{2}]$

Step 3

Differentiate.

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Step 3.1

By the Sum Rule, the derivative of $3 - x^{5}$ with respect to $x$ is $\frac{d}{d x} [3] + \frac{d}{d x} [- x^{5}]$ .

${(6 x - 5)}^{2} (2 (3 - x^{5}) (\frac{d}{d x} [3] + \frac{d}{d x} [- x^{5}])) + {(3 - x^{5})}^{2} \frac{d}{d x} [{(6 x - 5)}^{2}]$

Step 3.2

Since $3$ is constant with respect to $x$ , the derivative of $3$ with respect to $x$ is $0$ .

${(6 x - 5)}^{2} (2 (3 - x^{5}) (0 + \frac{d}{d x} [- x^{5}])) + {(3 - x^{5})}^{2} \frac{d}{d x} [{(6 x - 5)}^{2}]$

Step 3.3

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

${(6 x - 5)}^{2} (2 (3 - x^{5}) \frac{d}{d x} [- x^{5}]) + {(3 - x^{5})}^{2} \frac{d}{d x} [{(6 x - 5)}^{2}]$

Step 3.4

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

${(6 x - 5)}^{2} (2 (3 - x^{5}) (- \frac{d}{d x} [x^{5}])) + {(3 - x^{5})}^{2} \frac{d}{d x} [{(6 x - 5)}^{2}]$

Step 3.5

Multiply $- 1$ by $2$ .

${(6 x - 5)}^{2} (- 2 (3 - x^{5}) \frac{d}{d x} [x^{5}]) + {(3 - x^{5})}^{2} \frac{d}{d x} [{(6 x - 5)}^{2}]$

Step 3.6

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

${(6 x - 5)}^{2} (- 2 (3 - x^{5}) (5 x^{4})) + {(3 - x^{5})}^{2} \frac{d}{d x} [{(6 x - 5)}^{2}]$

Step 3.7

Multiply $5$ by $- 2$ .

${(6 x - 5)}^{2} (- 10 (3 - x^{5}) x^{4}) + {(3 - x^{5})}^{2} \frac{d}{d x} [{(6 x - 5)}^{2}]$

Step 4

Simplify.

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Step 4.1

Apply the distributive property.

${(6 x - 5)}^{2} ((- 10 \cdot 3 - 10 (- x^{5})) x^{4}) + {(3 - x^{5})}^{2} \frac{d}{d x} [{(6 x - 5)}^{2}]$

Step 4.2

Apply the distributive property.

${(6 x - 5)}^{2} (- 10 \cdot 3 x^{4} - 10 (- x^{5}) x^{4}) + {(3 - x^{5})}^{2} \frac{d}{d x} [{(6 x - 5)}^{2}]$

Step 4.3

Combine terms.

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Step 4.3.1

Multiply $- 10$ by $3$ .

${(6 x - 5)}^{2} (- 30 x^{4} - 10 (- x^{5}) x^{4}) + {(3 - x^{5})}^{2} \frac{d}{d x} [{(6 x - 5)}^{2}]$

Step 4.3.2

Multiply $- 1$ by $- 10$ .

${(6 x - 5)}^{2} (- 30 x^{4} + 10 x^{5} x^{4}) + {(3 - x^{5})}^{2} \frac{d}{d x} [{(6 x - 5)}^{2}]$

Step 4.3.3

Multiply $x^{5}$ by $x^{4}$ by adding the exponents.

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Step 4.3.3.1

Move $x^{4}$ .

${(6 x - 5)}^{2} (- 30 x^{4} + 10 (x^{4} x^{5})) + {(3 - x^{5})}^{2} \frac{d}{d x} [{(6 x - 5)}^{2}]$

Step 4.3.3.2

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

${(6 x - 5)}^{2} (- 30 x^{4} + 10 x^{4 + 5}) + {(3 - x^{5})}^{2} \frac{d}{d x} [{(6 x - 5)}^{2}]$

Step 4.3.3.3

Add $4$ and $5$ .

${(6 x - 5)}^{2} (- 30 x^{4} + 10 x^{9}) + {(3 - x^{5})}^{2} \frac{d}{d x} [{(6 x - 5)}^{2}]$

Step 4.4

Reorder terms.

${(6 x - 5)}^{2} (10 x^{9} - 30 x^{4}) + {(3 - x^{5})}^{2} \frac{d}{d x} [{(6 x - 5)}^{2}]$

Step 5

Rewrite ${(6 x - 5)}^{2}$ as $(6 x - 5) (6 x - 5)$ .

${(6 x - 5)}^{2} (10 x^{9} - 30 x^{4}) + {(3 - x^{5})}^{2} \frac{d}{d x} [(6 x - 5) (6 x - 5)]$

Step 6

Expand $(6 x - 5) (6 x - 5)$ using the FOIL Method.

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Step 6.1

Apply the distributive property.

${(6 x - 5)}^{2} (10 x^{9} - 30 x^{4}) + {(3 - x^{5})}^{2} \frac{d}{d x} [6 x (6 x - 5) - 5 (6 x - 5)]$

Step 6.2

Apply the distributive property.

${(6 x - 5)}^{2} (10 x^{9} - 30 x^{4}) + {(3 - x^{5})}^{2} \frac{d}{d x} [6 x (6 x) + 6 x \cdot - 5 - 5 (6 x - 5)]$

Step 6.3

Apply the distributive property.

${(6 x - 5)}^{2} (10 x^{9} - 30 x^{4}) + {(3 - x^{5})}^{2} \frac{d}{d x} [6 x (6 x) + 6 x \cdot - 5 - 5 (6 x) - 5 \cdot - 5]$

Step 7

Simplify and combine like terms.

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Step 7.1

Simplify each term.

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Step 7.1.1

Rewrite using the commutative property of multiplication.

${(6 x - 5)}^{2} (10 x^{9} - 30 x^{4}) + {(3 - x^{5})}^{2} \frac{d}{d x} [6 \cdot 6 x \cdot x + 6 x \cdot - 5 - 5 (6 x) - 5 \cdot - 5]$

Step 7.1.2

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

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Step 7.1.2.1

Move $x$ .

${(6 x - 5)}^{2} (10 x^{9} - 30 x^{4}) + {(3 - x^{5})}^{2} \frac{d}{d x} [6 \cdot 6 (x \cdot x) + 6 x \cdot - 5 - 5 (6 x) - 5 \cdot - 5]$

Step 7.1.2.2

Multiply $x$ by $x$ .

${(6 x - 5)}^{2} (10 x^{9} - 30 x^{4}) + {(3 - x^{5})}^{2} \frac{d}{d x} [6 \cdot 6 x^{2} + 6 x \cdot - 5 - 5 (6 x) - 5 \cdot - 5]$

Step 7.1.3

Multiply $6$ by $6$ .

${(6 x - 5)}^{2} (10 x^{9} - 30 x^{4}) + {(3 - x^{5})}^{2} \frac{d}{d x} [36 x^{2} + 6 x \cdot - 5 - 5 (6 x) - 5 \cdot - 5]$

Step 7.1.4

Multiply $- 5$ by $6$ .

${(6 x - 5)}^{2} (10 x^{9} - 30 x^{4}) + {(3 - x^{5})}^{2} \frac{d}{d x} [36 x^{2} - 30 x - 5 (6 x) - 5 \cdot - 5]$

Step 7.1.5

Multiply $6$ by $- 5$ .

${(6 x - 5)}^{2} (10 x^{9} - 30 x^{4}) + {(3 - x^{5})}^{2} \frac{d}{d x} [36 x^{2} - 30 x - 30 x - 5 \cdot - 5]$

Step 7.1.6

Multiply $- 5$ by $- 5$ .

${(6 x - 5)}^{2} (10 x^{9} - 30 x^{4}) + {(3 - x^{5})}^{2} \frac{d}{d x} [36 x^{2} - 30 x - 30 x + 25]$

Step 7.2

Subtract $30 x$ from $- 30 x$ .

${(6 x - 5)}^{2} (10 x^{9} - 30 x^{4}) + {(3 - x^{5})}^{2} \frac{d}{d x} [36 x^{2} - 60 x + 25]$

Step 8

By the Sum Rule, the derivative of $36 x^{2} - 60 x + 25$ with respect to $x$ is $\frac{d}{d x} [36 x^{2}] + \frac{d}{d x} [- 60 x] + \frac{d}{d x} [25]$ .

${(6 x - 5)}^{2} (10 x^{9} - 30 x^{4}) + {(3 - x^{5})}^{2} (\frac{d}{d x} [36 x^{2}] + \frac{d}{d x} [- 60 x] + \frac{d}{d x} [25])$

Step 9

Since $36$ is constant with respect to $x$ , the derivative of $36 x^{2}$ with respect to $x$ is $36 \frac{d}{d x} [x^{2}]$ .

${(6 x - 5)}^{2} (10 x^{9} - 30 x^{4}) + {(3 - x^{5})}^{2} (36 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 60 x] + \frac{d}{d x} [25])$

Step 10

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

${(6 x - 5)}^{2} (10 x^{9} - 30 x^{4}) + {(3 - x^{5})}^{2} (36 (2 x) + \frac{d}{d x} [- 60 x] + \frac{d}{d x} [25])$

Step 11

Multiply $2$ by $36$ .

${(6 x - 5)}^{2} (10 x^{9} - 30 x^{4}) + {(3 - x^{5})}^{2} (72 x + \frac{d}{d x} [- 60 x] + \frac{d}{d x} [25])$

Step 12

Since $- 60$ is constant with respect to $x$ , the derivative of $- 60 x$ with respect to $x$ is $- 60 \frac{d}{d x} [x]$ .

${(6 x - 5)}^{2} (10 x^{9} - 30 x^{4}) + {(3 - x^{5})}^{2} (72 x - 60 \frac{d}{d x} [x] + \frac{d}{d x} [25])$

Step 13

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

${(6 x - 5)}^{2} (10 x^{9} - 30 x^{4}) + {(3 - x^{5})}^{2} (72 x - 60 \cdot 1 + \frac{d}{d x} [25])$

Step 14

Multiply $- 60$ by $1$ .

${(6 x - 5)}^{2} (10 x^{9} - 30 x^{4}) + {(3 - x^{5})}^{2} (72 x - 60 + \frac{d}{d x} [25])$

Step 15

Since $25$ is constant with respect to $x$ , the derivative of $25$ with respect to $x$ is $0$ .

${(6 x - 5)}^{2} (10 x^{9} - 30 x^{4}) + {(3 - x^{5})}^{2} (72 x - 60 + 0)$

Step 16

Add $72 x - 60$ and $0$ .

${(6 x - 5)}^{2} (10 x^{9} - 30 x^{4}) + {(3 - x^{5})}^{2} (72 x - 60)$