Calculus Examples

$y = {(\sin^{5} (x^{3} + 7))}^{9}$

Step 1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{9}$ and $g (x) = \sin^{5} (x^{3} + 7)$ .

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Step 1.1

To apply the Chain Rule, set $u_{1}$ as $\sin^{5} (x^{3} + 7)$ .

$\frac{d}{d u_{1}} [{u_{1}}^{9}] \frac{d}{d x} [\sin^{5} (x^{3} + 7)]$

Step 1.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{1}} [{u_{1}}^{n}]$ is $n {u_{1}}^{n - 1}$ where $n = 9$ .

$9 {u_{1}}^{8} \frac{d}{d x} [\sin^{5} (x^{3} + 7)]$

Step 1.3

Replace all occurrences of $u_{1}$ with $\sin^{5} (x^{3} + 7)$ .

$9 {(\sin^{5} (x^{3} + 7))}^{8} \frac{d}{d x} [\sin^{5} (x^{3} + 7)]$

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{5}$ and $g (x) = \sin (x^{3} + 7)$ .

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Step 2.1

To apply the Chain Rule, set $u_{2}$ as $\sin (x^{3} + 7)$ .

$9 {(\sin^{5} (x^{3} + 7))}^{8} (\frac{d}{d u_{2}} [{u_{2}}^{5}] \frac{d}{d x} [\sin (x^{3} + 7)])$

Step 2.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{2}} [{u_{2}}^{n}]$ is $n {u_{2}}^{n - 1}$ where $n = 5$ .

$9 {(\sin^{5} (x^{3} + 7))}^{8} (5 {u_{2}}^{4} \frac{d}{d x} [\sin (x^{3} + 7)])$

Step 2.3

Replace all occurrences of $u_{2}$ with $\sin (x^{3} + 7)$ .

$9 {(\sin^{5} (x^{3} + 7))}^{8} (5 \sin^{4} (x^{3} + 7) \frac{d}{d x} [\sin (x^{3} + 7)])$

Step 3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \sin (x)$ and $g (x) = x^{3} + 7$ .

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Step 3.1

To apply the Chain Rule, set $u_{3}$ as $x^{3} + 7$ .

$9 {(\sin^{5} (x^{3} + 7))}^{8} (5 \sin^{4} (x^{3} + 7) (\frac{d}{d u_{3}} [\sin (u_{3})] \frac{d}{d x} [x^{3} + 7]))$

Step 3.2

The derivative of $\sin (u_{3})$ with respect to $u_{3}$ is $\cos (u_{3})$ .

$9 {(\sin^{5} (x^{3} + 7))}^{8} (5 \sin^{4} (x^{3} + 7) (\cos (u_{3}) \frac{d}{d x} [x^{3} + 7]))$

Step 3.3

Replace all occurrences of $u_{3}$ with $x^{3} + 7$ .

$9 {(\sin^{5} (x^{3} + 7))}^{8} (5 \sin^{4} (x^{3} + 7) (\cos (x^{3} + 7) \frac{d}{d x} [x^{3} + 7]))$

Step 4

Differentiate.

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Step 4.1

By the Sum Rule, the derivative of $x^{3} + 7$ with respect to $x$ is $\frac{d}{d x} [x^{3}] + \frac{d}{d x} [7]$ .

$9 {(\sin^{5} (x^{3} + 7))}^{8} (5 \sin^{4} (x^{3} + 7) \cos (x^{3} + 7) (\frac{d}{d x} [x^{3}] + \frac{d}{d x} [7]))$

Step 4.2

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

$9 {(\sin^{5} (x^{3} + 7))}^{8} (5 \sin^{4} (x^{3} + 7) \cos (x^{3} + 7) (3 x^{2} + \frac{d}{d x} [7]))$

Step 4.3

Since $7$ is constant with respect to $x$ , the derivative of $7$ with respect to $x$ is $0$ .

$9 {(\sin^{5} (x^{3} + 7))}^{8} (5 \sin^{4} (x^{3} + 7) \cos (x^{3} + 7) (3 x^{2} + 0))$

Step 4.4

Simplify the expression.

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Step 4.4.1

Add $3 x^{2}$ and $0$ .

$9 {(\sin^{5} (x^{3} + 7))}^{8} (5 \sin^{4} (x^{3} + 7) \cos (x^{3} + 7) (3 x^{2}))$

Step 4.4.2

Multiply $3$ by $5$ .

$9 {(\sin^{5} (x^{3} + 7))}^{8} (15 \sin^{4} (x^{3} + 7) \cos (x^{3} + 7) x^{2})$

Step 4.4.3

Reorder the factors of $15 \sin^{4} (x^{3} + 7) \cos (x^{3} + 7) x^{2}$ .

$9 {(\sin^{5} (x^{3} + 7))}^{8} (15 x^{2} \sin^{4} (x^{3} + 7) \cos (x^{3} + 7))$

Step 5

Combine terms.

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Step 5.1

Multiply the exponents in ${(\sin^{5} (x^{3} + 7))}^{8}$ .

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Step 5.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$9 {\sin (x^{3} + 7)}^{5 \cdot 8} (15 x^{2} \sin^{4} (x^{3} + 7) \cos (x^{3} + 7))$

Step 5.1.2

Multiply $5$ by $8$ .

$9 \sin^{40} (x^{3} + 7) (15 x^{2} \sin^{4} (x^{3} + 7) \cos (x^{3} + 7))$

Step 5.2

Multiply $15$ by $9$ .

$135 \sin^{40} (x^{3} + 7) (x^{2} \sin^{4} (x^{3} + 7) \cos (x^{3} + 7))$

Step 5.3

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

$135 (x^{2} {\sin (x^{3} + 7)}^{40 + 4} \cos (x^{3} + 7))$

Step 5.4

Add $40$ and $4$ .

$135 x^{2} \sin^{44} (x^{3} + 7) \cos (x^{3} + 7)$