Calculus Examples

$\frac{d}{d x} \cdot (\sin^{2} (x) \log_{7} (\sin^{3} (x)))$

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 $f (x) = \sin^{2} (x)$ and $g (x) = \log_{7} (\sin^{3} (x))$ .

$\sin^{2} (x) \frac{d}{d x} [\log_{7} (\sin^{3} (x))] + \log_{7} (\sin^{3} (x)) \frac{d}{d x} [\sin^{2} (x)]$

Step 2

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

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Step 2.1

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

$\sin^{2} (x) (\frac{d}{d u_{1}} [\log_{7} (u_{1})] \frac{d}{d x} [\sin^{3} (x)]) + \log_{7} (\sin^{3} (x)) \frac{d}{d x} [\sin^{2} (x)]$

Step 2.2

The derivative of $\log_{7} (u_{1})$ with respect to $u_{1}$ is $\frac{1}{u_{1} \ln (7)}$ .

$\sin^{2} (x) (\frac{1}{u_{1} \ln (7)} \frac{d}{d x} [\sin^{3} (x)]) + \log_{7} (\sin^{3} (x)) \frac{d}{d x} [\sin^{2} (x)]$

Step 2.3

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

$\sin^{2} (x) (\frac{1}{\sin^{3} (x) \ln (7)} \frac{d}{d x} [\sin^{3} (x)]) + \log_{7} (\sin^{3} (x)) \frac{d}{d x} [\sin^{2} (x)]$

Step 3

Simplify terms.

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Step 3.1

Combine $\frac{1}{\sin^{3} (x) \ln (7)}$ and $\sin^{2} (x)$ .

$\frac{\sin^{2} (x)}{\sin^{3} (x) \ln (7)} \frac{d}{d x} [\sin^{3} (x)] + \log_{7} (\sin^{3} (x)) \frac{d}{d x} [\sin^{2} (x)]$

Step 3.2

Cancel the common factor of $\sin^{2} (x)$ and $\sin^{3} (x)$ .

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Step 3.2.1

Multiply by $1$ .

$\frac{\sin^{2} (x) \cdot 1}{\sin^{3} (x) \ln (7)} \frac{d}{d x} [\sin^{3} (x)] + \log_{7} (\sin^{3} (x)) \frac{d}{d x} [\sin^{2} (x)]$

Step 3.2.2

Cancel the common factors.

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Step 3.2.2.1

Factor $\sin^{2} (x)$ out of $\sin^{3} (x) \ln (7)$ .

$\frac{\sin^{2} (x) \cdot 1}{\sin^{2} (x) (\sin (x) \ln (7))} \frac{d}{d x} [\sin^{3} (x)] + \log_{7} (\sin^{3} (x)) \frac{d}{d x} [\sin^{2} (x)]$

Step 3.2.2.2

Cancel the common factor.

$\frac{\sin^{2} (x) \cdot 1}{\sin^{2} (x) (\sin (x) \ln (7))} \frac{d}{d x} [\sin^{3} (x)] + \log_{7} (\sin^{3} (x)) \frac{d}{d x} [\sin^{2} (x)]$

Step 3.2.2.3

Rewrite the expression.

$\frac{1}{\sin (x) \ln (7)} \frac{d}{d x} [\sin^{3} (x)] + \log_{7} (\sin^{3} (x)) \frac{d}{d x} [\sin^{2} (x)]$

Step 4

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

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Step 4.1

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

$\frac{1}{\sin (x) \ln (7)} (\frac{d}{d u_{2}} [{u_{2}}^{3}] \frac{d}{d x} [\sin (x)]) + \log_{7} (\sin^{3} (x)) \frac{d}{d x} [\sin^{2} (x)]$

Step 4.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{2}} [{u_{2}}^{n}]$ is $n {u_{2}}^{n - 1}$ where $n = 3$ .

$\frac{1}{\sin (x) \ln (7)} (3 {u_{2}}^{2} \frac{d}{d x} [\sin (x)]) + \log_{7} (\sin^{3} (x)) \frac{d}{d x} [\sin^{2} (x)]$

Step 4.3

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

$\frac{1}{\sin (x) \ln (7)} (3 \sin^{2} (x) \frac{d}{d x} [\sin (x)]) + \log_{7} (\sin^{3} (x)) \frac{d}{d x} [\sin^{2} (x)]$

Step 5

Simplify terms.

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Step 5.1

Combine $3$ and $\frac{1}{\sin (x) \ln (7)}$ .

$\frac{3}{\sin (x) \ln (7)} (\sin^{2} (x) \frac{d}{d x} [\sin (x)]) + \log_{7} (\sin^{3} (x)) \frac{d}{d x} [\sin^{2} (x)]$

Step 5.2

Combine $\sin^{2} (x)$ and $\frac{3}{\sin (x) \ln (7)}$ .

$\frac{\sin^{2} (x) \cdot 3}{\sin (x) \ln (7)} \frac{d}{d x} [\sin (x)] + \log_{7} (\sin^{3} (x)) \frac{d}{d x} [\sin^{2} (x)]$

Step 5.3

Move $3$ to the left of $\sin^{2} (x)$ .

$\frac{3 \cdot \sin^{2} (x)}{\sin (x) \ln (7)} \frac{d}{d x} [\sin (x)] + \log_{7} (\sin^{3} (x)) \frac{d}{d x} [\sin^{2} (x)]$

Step 5.4

Cancel the common factor of $\sin^{2} (x)$ and $\sin (x)$ .

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Step 5.4.1

Factor $\sin (x)$ out of $3 \sin^{2} (x)$ .

$\frac{\sin (x) (3 \sin (x))}{\sin (x) \ln (7)} \frac{d}{d x} [\sin (x)] + \log_{7} (\sin^{3} (x)) \frac{d}{d x} [\sin^{2} (x)]$

Step 5.4.2

Cancel the common factors.

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Step 5.4.2.1

Factor $\sin (x)$ out of $\sin (x) \ln (7)$ .

$\frac{\sin (x) (3 \sin (x))}{\sin (x) (\ln (7))} \frac{d}{d x} [\sin (x)] + \log_{7} (\sin^{3} (x)) \frac{d}{d x} [\sin^{2} (x)]$

Step 5.4.2.2

Cancel the common factor.

$\frac{\sin (x) (3 \sin (x))}{\sin (x) \ln (7)} \frac{d}{d x} [\sin (x)] + \log_{7} (\sin^{3} (x)) \frac{d}{d x} [\sin^{2} (x)]$

Step 5.4.2.3

Rewrite the expression.

$\frac{3 \sin (x)}{\ln (7)} \frac{d}{d x} [\sin (x)] + \log_{7} (\sin^{3} (x)) \frac{d}{d x} [\sin^{2} (x)]$

Step 6

The derivative of $\sin (x)$ with respect to $x$ is $\cos (x)$ .

$\frac{3 \sin (x)}{\ln (7)} \cos (x) + \log_{7} (\sin^{3} (x)) \frac{d}{d x} [\sin^{2} (x)]$

Step 7

Combine $\frac{3 \sin (x)}{\ln (7)}$ and $\cos (x)$ .

$\frac{3 \sin (x) \cos (x)}{\ln (7)} + \log_{7} (\sin^{3} (x)) \frac{d}{d x} [\sin^{2} (x)]$

Step 8

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 $g (x) = \sin (x)$ .

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Step 8.1

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

$\frac{3 \sin (x) \cos (x)}{\ln (7)} + \log_{7} (\sin^{3} (x)) (\frac{d}{d u_{3}} [{u_{3}}^{2}] \frac{d}{d x} [\sin (x)])$

Step 8.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{3}} [{u_{3}}^{n}]$ is $n {u_{3}}^{n - 1}$ where $n = 2$ .

$\frac{3 \sin (x) \cos (x)}{\ln (7)} + \log_{7} (\sin^{3} (x)) (2 u_{3} \frac{d}{d x} [\sin (x)])$

Step 8.3

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

$\frac{3 \sin (x) \cos (x)}{\ln (7)} + \log_{7} (\sin^{3} (x)) (2 \sin (x) \frac{d}{d x} [\sin (x)])$

Step 9

Move $2$ to the left of $\log_{7} (\sin^{3} (x))$ .

$\frac{3 \sin (x) \cos (x)}{\ln (7)} + 2 \cdot \log_{7} (\sin^{3} (x)) \sin (x) \frac{d}{d x} [\sin (x)]$

Step 10

The derivative of $\sin (x)$ with respect to $x$ is $\cos (x)$ .

$\frac{3 \sin (x) \cos (x)}{\ln (7)} + 2 \log_{7} (\sin^{3} (x)) \sin (x) \cos (x)$

Step 11

Simplify.

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Step 11.1

Reorder terms.

$2 \cos (x) \sin (x) \log_{7} (\sin^{3} (x)) + \frac{3 \sin (x) \cos (x)}{\ln (7)}$

Step 11.2

Simplify each term.

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Step 11.2.1

Reorder $2 \cos (x)$ and $\sin (x)$ .

$\sin (x) (2 \cos (x)) \log_{7} (\sin^{3} (x)) + \frac{3 \sin (x) \cos (x)}{\ln (7)}$

Step 11.2.2

Reorder $\sin (x)$ and $2$ .

$2 \cdot \sin (x) \cos (x) \log_{7} (\sin^{3} (x)) + \frac{3 \sin (x) \cos (x)}{\ln (7)}$

Step 11.2.3

Apply the sine double-angle identity.

$\sin (2 x) \log_{7} (\sin^{3} (x)) + \frac{3 \sin (x) \cos (x)}{\ln (7)}$