Calculus Examples

$f (x) = \log (- \frac{3}{2} \cdot \cos (3 x^{2}))$

Step 1

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

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Step 1.1

To apply the Chain Rule, set $u_{1}$ as $- \frac{3}{2} \cdot \cos (3 x^{2})$ .

$\frac{d}{d u_{1}} [\log (u_{1})] \frac{d}{d x} [- \frac{3}{2} \cdot \cos (3 x^{2})]$

Step 1.2

The derivative of $\log (u_{1})$ with respect to $u_{1}$ is $\frac{1}{u_{1} \ln (10)}$ .

$\frac{1}{u_{1} \ln (10)} \frac{d}{d x} [- \frac{3}{2} \cdot \cos (3 x^{2})]$

Step 1.3

Replace all occurrences of $u_{1}$ with $- \frac{3}{2} \cdot \cos (3 x^{2})$ .

$\frac{1}{- \frac{3}{2} \cdot \cos (3 x^{2}) \ln (10)} \frac{d}{d x} [- \frac{3}{2} \cdot \cos (3 x^{2})]$

Step 2

Simplify terms.

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Step 2.1

Combine $\cos (3 x^{2})$ and $\frac{3}{2}$ .

$\frac{1}{- \frac{\cos (3 x^{2}) \cdot 3}{2} \ln (10)} \frac{d}{d x} [- \frac{3}{2} \cdot \cos (3 x^{2})]$

Step 2.2

Combine $\ln (10)$ and $\frac{\cos (3 x^{2}) \cdot 3}{2}$ .

$\frac{1}{- \frac{\ln (10) (\cos (3 x^{2}) \cdot 3)}{2}} \frac{d}{d x} [- \frac{3}{2} \cdot \cos (3 x^{2})]$

Step 2.3

Reorder.

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Step 2.3.1

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

$\frac{1}{- \frac{\ln (10) (3 \cdot \cos (3 x^{2}))}{2}} \frac{d}{d x} [- \frac{3}{2} \cdot \cos (3 x^{2})]$

Step 2.3.2

Move $3$ to the left of $\ln (10)$ .

$\frac{1}{- \frac{3 \cdot \ln (10) \cos (3 x^{2})}{2}} \frac{d}{d x} [- \frac{3}{2} \cdot \cos (3 x^{2})]$

Step 2.4

Cancel the common factor of $1$ and $- 1$ .

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Step 2.4.1

Rewrite $1$ as $- 1 (- 1)$ .

$\frac{- 1 (- 1)}{- \frac{3 \ln (10) \cos (3 x^{2})}{2}} \frac{d}{d x} [- \frac{3}{2} \cdot \cos (3 x^{2})]$

Step 2.4.2

Move the negative in front of the fraction.

$- \frac{1}{\frac{3 \ln (10) \cos (3 x^{2})}{2}} \frac{d}{d x} [- \frac{3}{2} \cdot \cos (3 x^{2})]$

Step 3

Multiply by the reciprocal of the fraction to divide by $\frac{3 \ln (10) \cos (3 x^{2})}{2}$ .

$- (1 \frac{2}{3 \ln (10) \cos (3 x^{2})}) \frac{d}{d x} [- \frac{3}{2} \cdot \cos (3 x^{2})]$

Step 4

Differentiate using the Constant Multiple Rule.

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Step 4.1

Multiply $\frac{2}{3 \ln (10) \cos (3 x^{2})}$ by $1$ .

$- \frac{2}{3 \ln (10) \cos (3 x^{2})} \frac{d}{d x} [- \frac{3}{2} \cdot \cos (3 x^{2})]$

Step 4.2

Combine fractions.

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Step 4.2.1

Combine $\cos (3 x^{2})$ and $\frac{3}{2}$ .

$- \frac{2}{3 \ln (10) \cos (3 x^{2})} \frac{d}{d x} [- \frac{\cos (3 x^{2}) \cdot 3}{2}]$

Step 4.2.2

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

$- \frac{2}{3 \ln (10) \cos (3 x^{2})} \frac{d}{d x} [- \frac{3 \cdot \cos (3 x^{2})}{2}]$

Step 4.3

Since $- \frac{3}{2}$ is constant with respect to $x$ , the derivative of $- \frac{3 \cos (3 x^{2})}{2}$ with respect to $x$ is $- \frac{3}{2} \frac{d}{d x} [\cos (3 x^{2})]$ .

$- \frac{2}{3 \ln (10) \cos (3 x^{2})} (- \frac{3}{2} \frac{d}{d x} [\cos (3 x^{2})])$

Step 4.4

Simplify terms.

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Step 4.4.1

Multiply $- 1$ by $- 1$ .

$1 \frac{2}{3 \ln (10) \cos (3 x^{2})} (\frac{3}{2} \frac{d}{d x} [\cos (3 x^{2})])$

Step 4.4.2

Multiply $\frac{2}{3 \ln (10) \cos (3 x^{2})}$ by $1$ .

$\frac{2}{3 \ln (10) \cos (3 x^{2})} (\frac{3}{2} \frac{d}{d x} [\cos (3 x^{2})])$

Step 4.4.3

Multiply $\frac{3}{2}$ by $\frac{2}{3 \ln (10) \cos (3 x^{2})}$ .

$\frac{3 \cdot 2}{2 (3 \ln (10) \cos (3 x^{2}))} \frac{d}{d x} [\cos (3 x^{2})]$

Step 4.4.4

Multiply.

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Step 4.4.4.1

Multiply $3$ by $2$ .

$\frac{6}{2 (3 \ln (10) \cos (3 x^{2}))} \frac{d}{d x} [\cos (3 x^{2})]$

Step 4.4.4.2

Multiply $3$ by $2$ .

$\frac{6}{6 (\ln (10) \cos (3 x^{2}))} \frac{d}{d x} [\cos (3 x^{2})]$

Step 4.4.5

Cancel the common factor of $6$ .

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Step 4.4.5.1

Cancel the common factor.

$\frac{6}{6 \ln (10) \cos (3 x^{2})} \frac{d}{d x} [\cos (3 x^{2})]$

Step 4.4.5.2

Rewrite the expression.

$\frac{1}{\ln (10) \cos (3 x^{2})} \frac{d}{d x} [\cos (3 x^{2})]$

Step 5

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

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Step 5.1

To apply the Chain Rule, set $u_{2}$ as $3 x^{2}$ .

$\frac{1}{\ln (10) \cos (3 x^{2})} (\frac{d}{d u_{2}} [\cos (u_{2})] \frac{d}{d x} [3 x^{2}])$

Step 5.2

The derivative of $\cos (u_{2})$ with respect to $u_{2}$ is $- \sin (u_{2})$ .

$\frac{1}{\ln (10) \cos (3 x^{2})} (- \sin (u_{2}) \frac{d}{d x} [3 x^{2}])$

Step 5.3

Replace all occurrences of $u_{2}$ with $3 x^{2}$ .

$\frac{1}{\ln (10) \cos (3 x^{2})} (- \sin (3 x^{2}) \frac{d}{d x} [3 x^{2}])$

Step 6

Differentiate.

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Step 6.1

Combine $\frac{1}{\ln (10) \cos (3 x^{2})}$ and $\sin (3 x^{2})$ .

$- \frac{\sin (3 x^{2})}{\ln (10) \cos (3 x^{2})} \frac{d}{d x} [3 x^{2}]$

Step 6.2

Since $3$ is constant with respect to $x$ , the derivative of $3 x^{2}$ with respect to $x$ is $3 \frac{d}{d x} [x^{2}]$ .

$- \frac{\sin (3 x^{2})}{\ln (10) \cos (3 x^{2})} (3 \frac{d}{d x} [x^{2}])$

Step 6.3

Combine fractions.

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Step 6.3.1

Multiply $3$ by $- 1$ .

$- 3 \frac{\sin (3 x^{2})}{\ln (10) \cos (3 x^{2})} \frac{d}{d x} [x^{2}]$

Step 6.3.2

Combine $- 3$ and $\frac{\sin (3 x^{2})}{\ln (10) \cos (3 x^{2})}$ .

$\frac{- 3 \sin (3 x^{2})}{\ln (10) \cos (3 x^{2})} \frac{d}{d x} [x^{2}]$

Step 6.3.3

Move the negative in front of the fraction.

$- \frac{3 \sin (3 x^{2})}{\ln (10) \cos (3 x^{2})} \frac{d}{d x} [x^{2}]$

Step 6.4

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

$- \frac{3 \sin (3 x^{2})}{\ln (10) \cos (3 x^{2})} (2 x)$

Step 6.5

Combine fractions.

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Step 6.5.1

Multiply $2$ by $- 1$ .

$- 2 \frac{3 \sin (3 x^{2})}{\ln (10) \cos (3 x^{2})} x$

Step 6.5.2

Combine $- 2$ and $\frac{3 \sin (3 x^{2})}{\ln (10) \cos (3 x^{2})}$ .

$\frac{- 2 (3 \sin (3 x^{2}))}{\ln (10) \cos (3 x^{2})} x$

Step 6.5.3

Multiply $3$ by $- 2$ .

$\frac{- 6 \sin (3 x^{2})}{\ln (10) \cos (3 x^{2})} x$

Step 6.5.4

Combine $\frac{- 6 \sin (3 x^{2})}{\ln (10) \cos (3 x^{2})}$ and $x$ .

$\frac{- 6 \sin (3 x^{2}) x}{\ln (10) \cos (3 x^{2})}$

Step 6.5.5

Move the negative in front of the fraction.

$- \frac{6 \sin (3 x^{2}) x}{\ln (10) \cos (3 x^{2})}$

Step 7

Simplify.

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Step 7.1

Reorder factors in $6 \sin (3 x^{2}) x$ .

$- \frac{6 x \sin (3 x^{2})}{\ln (10) \cos (3 x^{2})}$

Step 7.2

Separate fractions.

$- (\frac{6 x}{\ln (10)} \cdot \frac{\sin (3 x^{2})}{\cos (3 x^{2})})$

Step 7.3

Convert from $\frac{\sin (3 x^{2})}{\cos (3 x^{2})}$ to $\tan (3 x^{2})$ .

$- (\frac{6 x}{\ln (10)} \tan (3 x^{2}))$

Step 7.4

Combine $\frac{6 x}{\ln (10)}$ and $\tan (3 x^{2})$ .

$- \frac{6 x \tan (3 x^{2})}{\ln (10)}$