Calculus Examples

$f (x) = {(3 x^{5} + 5)}^{6}$

Step 1

Find the first derivative.

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Step 1.1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{6}$ and $g (x) = 3 x^{5} + 5$ .

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Step 1.1.1

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

$\frac{d}{d u} [u^{6}] \frac{d}{d x} [3 x^{5} + 5]$

Step 1.1.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 6$ .

$6 u^{5} \frac{d}{d x} [3 x^{5} + 5]$

Step 1.1.3

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

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

Step 1.2

Differentiate.

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Step 1.2.1

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

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

Step 1.2.2

Since $3$ is constant with respect to $x$ , the derivative of $3 x^{5}$ with respect to $x$ is $3 \frac{d}{d x} [x^{5}]$ .

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

Step 1.2.3

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

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

Step 1.2.4

Multiply $5$ by $3$ .

$6 {(3 x^{5} + 5)}^{5} (15 x^{4} + \frac{d}{d x} [5])$

Step 1.2.5

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

$6 {(3 x^{5} + 5)}^{5} (15 x^{4} + 0)$

Step 1.2.6

Simplify the expression.

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Step 1.2.6.1

Add $15 x^{4}$ and $0$ .

$6 {(3 x^{5} + 5)}^{5} (15 x^{4})$

Step 1.2.6.2

Multiply $15$ by $6$ .

$90 {(3 x^{5} + 5)}^{5} x^{4}$

Step 1.2.6.3

Reorder the factors of $90 {(3 x^{5} + 5)}^{5} x^{4}$ .

$f' (x) = 90 x^{4} {(3 x^{5} + 5)}^{5}$

Step 2

Find the second derivative.

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Step 2.1

Since $90$ is constant with respect to $x$ , the derivative of $90 x^{4} {(3 x^{5} + 5)}^{5}$ with respect to $x$ is $90 \frac{d}{d x} [x^{4} {(3 x^{5} + 5)}^{5}]$ .

$90 \frac{d}{d x} [x^{4} {(3 x^{5} + 5)}^{5}]$

Step 2.2

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

$90 (x^{4} \frac{d}{d x} [{(3 x^{5} + 5)}^{5}] + {(3 x^{5} + 5)}^{5} \frac{d}{d x} [x^{4}])$

Step 2.3

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

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Step 2.3.1

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

$90 (x^{4} (\frac{d}{d u} [u^{5}] \frac{d}{d x} [3 x^{5} + 5]) + {(3 x^{5} + 5)}^{5} \frac{d}{d x} [x^{4}])$

Step 2.3.2

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

$90 (x^{4} (5 u^{4} \frac{d}{d x} [3 x^{5} + 5]) + {(3 x^{5} + 5)}^{5} \frac{d}{d x} [x^{4}])$

Step 2.3.3

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

$90 (x^{4} (5 {(3 x^{5} + 5)}^{4} \frac{d}{d x} [3 x^{5} + 5]) + {(3 x^{5} + 5)}^{5} \frac{d}{d x} [x^{4}])$

Step 2.4

Differentiate.

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Step 2.4.1

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

$90 (x^{4} (5 {(3 x^{5} + 5)}^{4} (\frac{d}{d x} [3 x^{5}] + \frac{d}{d x} [5])) + {(3 x^{5} + 5)}^{5} \frac{d}{d x} [x^{4}])$

Step 2.4.2

Since $3$ is constant with respect to $x$ , the derivative of $3 x^{5}$ with respect to $x$ is $3 \frac{d}{d x} [x^{5}]$ .

$90 (x^{4} (5 {(3 x^{5} + 5)}^{4} (3 \frac{d}{d x} [x^{5}] + \frac{d}{d x} [5])) + {(3 x^{5} + 5)}^{5} \frac{d}{d x} [x^{4}])$

Step 2.4.3

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

$90 (x^{4} (5 {(3 x^{5} + 5)}^{4} (3 (5 x^{4}) + \frac{d}{d x} [5])) + {(3 x^{5} + 5)}^{5} \frac{d}{d x} [x^{4}])$

Step 2.4.4

Multiply $5$ by $3$ .

$90 (x^{4} (5 {(3 x^{5} + 5)}^{4} (15 x^{4} + \frac{d}{d x} [5])) + {(3 x^{5} + 5)}^{5} \frac{d}{d x} [x^{4}])$

Step 2.4.5

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

$90 (x^{4} (5 {(3 x^{5} + 5)}^{4} (15 x^{4} + 0)) + {(3 x^{5} + 5)}^{5} \frac{d}{d x} [x^{4}])$

Step 2.4.6

Simplify the expression.

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Step 2.4.6.1

Add $15 x^{4}$ and $0$ .

$90 (x^{4} (5 {(3 x^{5} + 5)}^{4} (15 x^{4})) + {(3 x^{5} + 5)}^{5} \frac{d}{d x} [x^{4}])$

Step 2.4.6.2

Multiply $15$ by $5$ .

$90 (x^{4} (75 {(3 x^{5} + 5)}^{4} x^{4}) + {(3 x^{5} + 5)}^{5} \frac{d}{d x} [x^{4}])$

Step 2.5

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

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Step 2.5.1

Move $x^{4}$ .

$90 (x^{4} x^{4} (75 {(3 x^{5} + 5)}^{4}) + {(3 x^{5} + 5)}^{5} \frac{d}{d x} [x^{4}])$

Step 2.5.2

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

$90 (x^{4 + 4} (75 {(3 x^{5} + 5)}^{4}) + {(3 x^{5} + 5)}^{5} \frac{d}{d x} [x^{4}])$

Step 2.5.3

Add $4$ and $4$ .

$90 (x^{8} (75 {(3 x^{5} + 5)}^{4}) + {(3 x^{5} + 5)}^{5} \frac{d}{d x} [x^{4}])$

Step 2.6

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

$90 (x^{8} (75 {(3 x^{5} + 5)}^{4}) + {(3 x^{5} + 5)}^{5} (4 x^{3}))$

Step 2.7

Move $4$ to the left of ${(3 x^{5} + 5)}^{5}$ .

$90 (x^{8} (75 {(3 x^{5} + 5)}^{4}) + 4 \cdot {(3 x^{5} + 5)}^{5} x^{3})$

Step 2.8

Simplify.

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Step 2.8.1

Apply the distributive property.

$90 (x^{8} (75 {(3 x^{5} + 5)}^{4})) + 90 (4 {(3 x^{5} + 5)}^{5} x^{3})$

Step 2.8.2

Multiply $75$ by $90$ .

$6750 (x^{8} ({(3 x^{5} + 5)}^{4})) + 90 (4 {(3 x^{5} + 5)}^{5} x^{3})$

Step 2.8.3

Multiply $4$ by $90$ .

$6750 x^{8} {(3 x^{5} + 5)}^{4} + 360 ({(3 x^{5} + 5)}^{5} x^{3})$

Step 2.8.4

Factor $90 x^{3} {(3 x^{5} + 5)}^{4}$ out of $6750 x^{8} {(3 x^{5} + 5)}^{4} + 360 {(3 x^{5} + 5)}^{5} x^{3}$ .

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Step 2.8.4.1

Factor $90 x^{3} {(3 x^{5} + 5)}^{4}$ out of $6750 x^{8} {(3 x^{5} + 5)}^{4}$ .

$90 x^{3} {(3 x^{5} + 5)}^{4} (75 x^{5}) + 360 {(3 x^{5} + 5)}^{5} x^{3}$

Step 2.8.4.2

Factor $90 x^{3} {(3 x^{5} + 5)}^{4}$ out of $360 {(3 x^{5} + 5)}^{5} x^{3}$ .

$90 x^{3} {(3 x^{5} + 5)}^{4} (75 x^{5}) + 90 x^{3} {(3 x^{5} + 5)}^{4} (4 (3 x^{5} + 5))$

Step 2.8.4.3

Factor $90 x^{3} {(3 x^{5} + 5)}^{4}$ out of $90 x^{3} {(3 x^{5} + 5)}^{4} (75 x^{5}) + 90 x^{3} {(3 x^{5} + 5)}^{4} (4 (3 x^{5} + 5))$ .

$f'' (x) = 90 x^{3} {(3 x^{5} + 5)}^{4} (75 x^{5} + 4 (3 x^{5} + 5))$

Step 3

The second derivative of $f (x)$ with respect to $x$ is $90 x^{3} {(3 x^{5} + 5)}^{4} (75 x^{5} + 4 (3 x^{5} + 5))$ .

$90 x^{3} {(3 x^{5} + 5)}^{4} (75 x^{5} + 4 (3 x^{5} + 5))$