Calculus Examples

$y = 14 x (\sin (\ln (14 x)) + \cos (\ln (14 x)))$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (14 x (\sin (\ln (14 x)) + \cos (\ln (14 x))))$

Step 2

The derivative of $y$ with respect to $x$ is $y'$ .

$y'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

Since $14$ is constant with respect to $x$ , the derivative of $14 x (\sin (\ln (14 x)) + \cos (\ln (14 x)))$ with respect to $x$ is $14 \frac{d}{d x} [x (\sin (\ln (14 x)) + \cos (\ln (14 x)))]$ .

$14 \frac{d}{d x} [x (\sin (\ln (14 x)) + \cos (\ln (14 x)))]$

Step 3.2

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 $f (x) = x$ and $g (x) = \sin (\ln (14 x)) + \cos (\ln (14 x))$ .

$14 (x \frac{d}{d x} [\sin (\ln (14 x)) + \cos (\ln (14 x))] + (\sin (\ln (14 x)) + \cos (\ln (14 x))) \frac{d}{d x} [x])$

Step 3.3

By the Sum Rule, the derivative of $\sin (\ln (14 x)) + \cos (\ln (14 x))$ with respect to $x$ is $\frac{d}{d x} [\sin (\ln (14 x))] + \frac{d}{d x} [\cos (\ln (14 x))]$ .

$14 (x (\frac{d}{d x} [\sin (\ln (14 x))] + \frac{d}{d x} [\cos (\ln (14 x))]) + (\sin (\ln (14 x)) + \cos (\ln (14 x))) \frac{d}{d x} [x])$

Step 3.4

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \sin (x)$ and $g (x) = \ln (14 x)$ .

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Step 3.4.1

To apply the Chain Rule, set $u_{1}$ as $\ln (14 x)$ .

$14 (x (\frac{d}{d u_{1}} [\sin (u_{1})] \frac{d}{d x} [\ln (14 x)] + \frac{d}{d x} [\cos (\ln (14 x))]) + (\sin (\ln (14 x)) + \cos (\ln (14 x))) \frac{d}{d x} [x])$

Step 3.4.2

The derivative of $\sin (u_{1})$ with respect to $u_{1}$ is $\cos (u_{1})$ .

$14 (x (\cos (u_{1}) \frac{d}{d x} [\ln (14 x)] + \frac{d}{d x} [\cos (\ln (14 x))]) + (\sin (\ln (14 x)) + \cos (\ln (14 x))) \frac{d}{d x} [x])$

Step 3.4.3

Replace all occurrences of $u_{1}$ with $\ln (14 x)$ .

$14 (x (\cos (\ln (14 x)) \frac{d}{d x} [\ln (14 x)] + \frac{d}{d x} [\cos (\ln (14 x))]) + (\sin (\ln (14 x)) + \cos (\ln (14 x))) \frac{d}{d x} [x])$

Step 3.5

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \ln (x)$ and $g (x) = 14 x$ .

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Step 3.5.1

To apply the Chain Rule, set $u_{2}$ as $14 x$ .

$14 (x (\cos (\ln (14 x)) (\frac{d}{d u_{2}} [\ln (u_{2})] \frac{d}{d x} [14 x]) + \frac{d}{d x} [\cos (\ln (14 x))]) + (\sin (\ln (14 x)) + \cos (\ln (14 x))) \frac{d}{d x} [x])$

Step 3.5.2

The derivative of $\ln (u_{2})$ with respect to $u_{2}$ is $\frac{1}{u_{2}}$ .

$14 (x (\cos (\ln (14 x)) (\frac{1}{u_{2}} \frac{d}{d x} [14 x]) + \frac{d}{d x} [\cos (\ln (14 x))]) + (\sin (\ln (14 x)) + \cos (\ln (14 x))) \frac{d}{d x} [x])$

Step 3.5.3

Replace all occurrences of $u_{2}$ with $14 x$ .

$14 (x (\cos (\ln (14 x)) (\frac{1}{14 x} \frac{d}{d x} [14 x]) + \frac{d}{d x} [\cos (\ln (14 x))]) + (\sin (\ln (14 x)) + \cos (\ln (14 x))) \frac{d}{d x} [x])$

Step 3.6

Differentiate.

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Step 3.6.1

Combine $\frac{1}{14 x}$ and $\cos (\ln (14 x))$ .

$14 (x (\frac{\cos (\ln (14 x))}{14 x} \frac{d}{d x} [14 x] + \frac{d}{d x} [\cos (\ln (14 x))]) + (\sin (\ln (14 x)) + \cos (\ln (14 x))) \frac{d}{d x} [x])$

Step 3.6.2

Since $14$ is constant with respect to $x$ , the derivative of $14 x$ with respect to $x$ is $14 \frac{d}{d x} [x]$ .

$14 (x (\frac{\cos (\ln (14 x))}{14 x} (14 \frac{d}{d x} [x]) + \frac{d}{d x} [\cos (\ln (14 x))]) + (\sin (\ln (14 x)) + \cos (\ln (14 x))) \frac{d}{d x} [x])$

Step 3.6.3

Simplify terms.

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Step 3.6.3.1

Combine $14$ and $\frac{\cos (\ln (14 x))}{14 x}$ .

$14 (x (\frac{14 \cos (\ln (14 x))}{14 x} \frac{d}{d x} [x] + \frac{d}{d x} [\cos (\ln (14 x))]) + (\sin (\ln (14 x)) + \cos (\ln (14 x))) \frac{d}{d x} [x])$

Step 3.6.3.2

Cancel the common factor of $14$ .

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Step 3.6.3.2.1

Cancel the common factor.

$14 (x (\frac{14 \cos (\ln (14 x))}{14 x} \frac{d}{d x} [x] + \frac{d}{d x} [\cos (\ln (14 x))]) + (\sin (\ln (14 x)) + \cos (\ln (14 x))) \frac{d}{d x} [x])$

Step 3.6.3.2.2

Rewrite the expression.

$14 (x (\frac{\cos (\ln (14 x))}{x} \frac{d}{d x} [x] + \frac{d}{d x} [\cos (\ln (14 x))]) + (\sin (\ln (14 x)) + \cos (\ln (14 x))) \frac{d}{d x} [x])$

Step 3.6.4

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

$14 (x (\frac{\cos (\ln (14 x))}{x} \cdot 1 + \frac{d}{d x} [\cos (\ln (14 x))]) + (\sin (\ln (14 x)) + \cos (\ln (14 x))) \frac{d}{d x} [x])$

Step 3.6.5

Multiply $\frac{\cos (\ln (14 x))}{x}$ by $1$ .

$14 (x (\frac{\cos (\ln (14 x))}{x} + \frac{d}{d x} [\cos (\ln (14 x))]) + (\sin (\ln (14 x)) + \cos (\ln (14 x))) \frac{d}{d x} [x])$

Step 3.7

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 $g (x) = \ln (14 x)$ .

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Step 3.7.1

To apply the Chain Rule, set $u_{3}$ as $\ln (14 x)$ .

$14 (x (\frac{\cos (\ln (14 x))}{x} + \frac{d}{d u_{3}} [\cos (u_{3})] \frac{d}{d x} [\ln (14 x)]) + (\sin (\ln (14 x)) + \cos (\ln (14 x))) \frac{d}{d x} [x])$

Step 3.7.2

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

$14 (x (\frac{\cos (\ln (14 x))}{x} - \sin (u_{3}) \frac{d}{d x} [\ln (14 x)]) + (\sin (\ln (14 x)) + \cos (\ln (14 x))) \frac{d}{d x} [x])$

Step 3.7.3

Replace all occurrences of $u_{3}$ with $\ln (14 x)$ .

$14 (x (\frac{\cos (\ln (14 x))}{x} - \sin (\ln (14 x)) \frac{d}{d x} [\ln (14 x)]) + (\sin (\ln (14 x)) + \cos (\ln (14 x))) \frac{d}{d x} [x])$

Step 3.8

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \ln (x)$ and $g (x) = 14 x$ .

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Step 3.8.1

To apply the Chain Rule, set $u_{4}$ as $14 x$ .

$14 (x (\frac{\cos (\ln (14 x))}{x} - \sin (\ln (14 x)) (\frac{d}{d u_{4}} [\ln (u_{4})] \frac{d}{d x} [14 x])) + (\sin (\ln (14 x)) + \cos (\ln (14 x))) \frac{d}{d x} [x])$

Step 3.8.2

The derivative of $\ln (u_{4})$ with respect to $u_{4}$ is $\frac{1}{u_{4}}$ .

$14 (x (\frac{\cos (\ln (14 x))}{x} - \sin (\ln (14 x)) (\frac{1}{u_{4}} \frac{d}{d x} [14 x])) + (\sin (\ln (14 x)) + \cos (\ln (14 x))) \frac{d}{d x} [x])$

Step 3.8.3

Replace all occurrences of $u_{4}$ with $14 x$ .

$14 (x (\frac{\cos (\ln (14 x))}{x} - \sin (\ln (14 x)) (\frac{1}{14 x} \frac{d}{d x} [14 x])) + (\sin (\ln (14 x)) + \cos (\ln (14 x))) \frac{d}{d x} [x])$

Step 3.9

Differentiate.

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Step 3.9.1

Combine $\frac{1}{14 x}$ and $\sin (\ln (14 x))$ .

$14 (x (\frac{\cos (\ln (14 x))}{x} - \frac{\sin (\ln (14 x))}{14 x} \frac{d}{d x} [14 x]) + (\sin (\ln (14 x)) + \cos (\ln (14 x))) \frac{d}{d x} [x])$

Step 3.9.2

Since $14$ is constant with respect to $x$ , the derivative of $14 x$ with respect to $x$ is $14 \frac{d}{d x} [x]$ .

$14 (x (\frac{\cos (\ln (14 x))}{x} - \frac{\sin (\ln (14 x))}{14 x} (14 \frac{d}{d x} [x])) + (\sin (\ln (14 x)) + \cos (\ln (14 x))) \frac{d}{d x} [x])$

Step 3.9.3

Simplify terms.

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Step 3.9.3.1

Multiply $14$ by $- 1$ .

$14 (x (\frac{\cos (\ln (14 x))}{x} - 14 \frac{\sin (\ln (14 x))}{14 x} \frac{d}{d x} [x]) + (\sin (\ln (14 x)) + \cos (\ln (14 x))) \frac{d}{d x} [x])$

Step 3.9.3.2

Combine $- 14$ and $\frac{\sin (\ln (14 x))}{14 x}$ .

$14 (x (\frac{\cos (\ln (14 x))}{x} + \frac{- 14 \sin (\ln (14 x))}{14 x} \frac{d}{d x} [x]) + (\sin (\ln (14 x)) + \cos (\ln (14 x))) \frac{d}{d x} [x])$

Step 3.9.3.3

Cancel the common factor of $- 14$ and $14$ .

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Step 3.9.3.3.1

Factor $14$ out of $- 14 \sin (\ln (14 x))$ .

$14 (x (\frac{\cos (\ln (14 x))}{x} + \frac{14 (- \sin (\ln (14 x)))}{14 x} \frac{d}{d x} [x]) + (\sin (\ln (14 x)) + \cos (\ln (14 x))) \frac{d}{d x} [x])$

Step 3.9.3.3.2

Cancel the common factors.

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Step 3.9.3.3.2.1

Factor $14$ out of $14 x$ .

$14 (x (\frac{\cos (\ln (14 x))}{x} + \frac{14 (- \sin (\ln (14 x)))}{14 (x)} \frac{d}{d x} [x]) + (\sin (\ln (14 x)) + \cos (\ln (14 x))) \frac{d}{d x} [x])$

Step 3.9.3.3.2.2

Cancel the common factor.

$14 (x (\frac{\cos (\ln (14 x))}{x} + \frac{14 (- \sin (\ln (14 x)))}{14 x} \frac{d}{d x} [x]) + (\sin (\ln (14 x)) + \cos (\ln (14 x))) \frac{d}{d x} [x])$

Step 3.9.3.3.2.3

Rewrite the expression.

$14 (x (\frac{\cos (\ln (14 x))}{x} + \frac{- \sin (\ln (14 x))}{x} \frac{d}{d x} [x]) + (\sin (\ln (14 x)) + \cos (\ln (14 x))) \frac{d}{d x} [x])$

Step 3.9.3.4

Move the negative in front of the fraction.

$14 (x (\frac{\cos (\ln (14 x))}{x} - \frac{\sin (\ln (14 x))}{x} \frac{d}{d x} [x]) + (\sin (\ln (14 x)) + \cos (\ln (14 x))) \frac{d}{d x} [x])$

Step 3.9.4

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

$14 (x (\frac{\cos (\ln (14 x))}{x} - \frac{\sin (\ln (14 x))}{x} \cdot 1) + (\sin (\ln (14 x)) + \cos (\ln (14 x))) \frac{d}{d x} [x])$

Step 3.9.5

Multiply $- 1$ by $1$ .

$14 (x (\frac{\cos (\ln (14 x))}{x} - \frac{\sin (\ln (14 x))}{x}) + (\sin (\ln (14 x)) + \cos (\ln (14 x))) \frac{d}{d x} [x])$

Step 3.9.6

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

$14 (x (\frac{\cos (\ln (14 x))}{x} - \frac{\sin (\ln (14 x))}{x}) + (\sin (\ln (14 x)) + \cos (\ln (14 x))) \cdot 1)$

Step 3.9.7

Multiply $\sin (\ln (14 x)) + \cos (\ln (14 x))$ by $1$ .

$14 (x (\frac{\cos (\ln (14 x))}{x} - \frac{\sin (\ln (14 x))}{x}) + \sin (\ln (14 x)) + \cos (\ln (14 x)))$

Step 3.10

Simplify.

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Step 3.10.1

Apply the distributive property.

$14 (x \frac{\cos (\ln (14 x))}{x} + x (- \frac{\sin (\ln (14 x))}{x}) + \sin (\ln (14 x)) + \cos (\ln (14 x)))$

Step 3.10.2

Apply the distributive property.

$14 (x \frac{\cos (\ln (14 x))}{x}) + 14 (x (- \frac{\sin (\ln (14 x))}{x})) + 14 \sin (\ln (14 x)) + 14 \cos (\ln (14 x))$

Step 3.10.3

Combine terms.

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Step 3.10.3.1

Combine $x$ and $\frac{\cos (\ln (14 x))}{x}$ .

$14 \frac{x \cos (\ln (14 x))}{x} + 14 (x (- \frac{\sin (\ln (14 x))}{x})) + 14 \sin (\ln (14 x)) + 14 \cos (\ln (14 x))$

Step 3.10.3.2

Cancel the common factor of $x$ .

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Step 3.10.3.2.1

Cancel the common factor.

$14 \frac{x \cos (\ln (14 x))}{x} + 14 (x (- \frac{\sin (\ln (14 x))}{x})) + 14 \sin (\ln (14 x)) + 14 \cos (\ln (14 x))$

Step 3.10.3.2.2

Divide $\cos (\ln (14 x))$ by $1$ .

$14 \cos (\ln (14 x)) + 14 (x (- \frac{\sin (\ln (14 x))}{x})) + 14 \sin (\ln (14 x)) + 14 \cos (\ln (14 x))$

Step 3.10.3.3

Combine $x$ and $\frac{\sin (\ln (14 x))}{x}$ .

$14 \cos (\ln (14 x)) + 14 (- \frac{x \sin (\ln (14 x))}{x}) + 14 \sin (\ln (14 x)) + 14 \cos (\ln (14 x))$

Step 3.10.3.4

Cancel the common factor of $x$ .

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Step 3.10.3.4.1

Cancel the common factor.

$14 \cos (\ln (14 x)) + 14 (- \frac{x \sin (\ln (14 x))}{x}) + 14 \sin (\ln (14 x)) + 14 \cos (\ln (14 x))$

Step 3.10.3.4.2

Divide $\sin (\ln (14 x))$ by $1$ .

$14 \cos (\ln (14 x)) + 14 (- \sin (\ln (14 x))) + 14 \sin (\ln (14 x)) + 14 \cos (\ln (14 x))$

Step 3.10.3.5

Multiply $- 1$ by $14$ .

$14 \cos (\ln (14 x)) - 14 \sin (\ln (14 x)) + 14 \sin (\ln (14 x)) + 14 \cos (\ln (14 x))$

Step 3.10.3.6

Add $- 14 \sin (\ln (14 x))$ and $14 \sin (\ln (14 x))$ .

$14 \cos (\ln (14 x)) + 0 + 14 \cos (\ln (14 x))$

Step 3.10.3.7

Add $14 \cos (\ln (14 x))$ and $0$ .

$14 \cos (\ln (14 x)) + 14 \cos (\ln (14 x))$

Step 3.10.3.8

Add $14 \cos (\ln (14 x))$ and $14 \cos (\ln (14 x))$ .

$28 \cos (\ln (14 x))$

Step 4

Reform the equation by setting the left side equal to the right side.

$y' = 28 \cos (\ln (14 x))$

Step 5

Replace $y'$ with $\frac{d y}{d x}$ .

$\frac{d y}{d x} = 28 \cos (\ln (14 x))$