Calculus Examples

$\cos (r) + \cot (x) = e^{r x}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (\cos (r) + \cot (x)) = \frac{d}{d x} (e^{r x})$

Step 2

Differentiate the left side of the equation.

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Step 2.1

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

$\frac{d}{d x} [\cos (r)] + \frac{d}{d x} [\cot (x)]$

Step 2.2

Evaluate $\frac{d}{d x} [\cos (r)]$ .

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Step 2.2.1

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

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Step 2.2.1.1

To apply the Chain Rule, set $u$ as $r$ .

$\frac{d}{d u} [\cos (u)] \frac{d}{d x} [r] + \frac{d}{d x} [\cot (x)]$

Step 2.2.1.2

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

$- \sin (u) \frac{d}{d x} [r] + \frac{d}{d x} [\cot (x)]$

Step 2.2.1.3

Replace all occurrences of $u$ with $r$ .

$- \sin (r) \frac{d}{d x} [r] + \frac{d}{d x} [\cot (x)]$

Step 2.2.2

Rewrite $\frac{d}{d x} [r]$ as $r'$ .

$- \sin (r) r' + \frac{d}{d x} [\cot (x)]$

Step 2.3

The derivative of $\cot (x)$ with respect to $x$ is $- \csc^{2} (x)$ .

$- \sin (r) r' - \csc^{2} (x)$

Step 3

Differentiate the right side of the equation.

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Step 3.1

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

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Step 3.1.1

To apply the Chain Rule, set $u$ as $r x$ .

$\frac{d}{d u} [e^{u}] \frac{d}{d x} [r x]$

Step 3.1.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u} [a^{u}]$ is $a^{u} \ln (a)$ where $a$ = $e$ .

$e^{u} \frac{d}{d x} [r x]$

Step 3.1.3

Replace all occurrences of $u$ with $r x$ .

$e^{r x} \frac{d}{d x} [r x]$

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 $f (x) = r$ and $g (x) = x$ .

$e^{r x} (r \frac{d}{d x} [x] + x \frac{d}{d x} [r])$

Step 3.3

Differentiate using the Power Rule.

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Step 3.3.1

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

$e^{r x} (r \cdot 1 + x \frac{d}{d x} [r])$

Step 3.3.2

Multiply $r$ by $1$ .

$e^{r x} (r + x \frac{d}{d x} [r])$

Step 3.4

Rewrite $\frac{d}{d x} [r]$ as $r'$ .

$e^{r x} (r + x r')$

Step 3.5

Simplify.

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Step 3.5.1

Apply the distributive property.

$e^{r x} r + e^{r x} (x r')$

Step 3.5.2

Reorder terms.

$e^{r x} r + e^{r x} x r'$

Step 4

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

$- \sin (r) r' - \csc^{2} (x) = e^{r x} r + e^{r x} x r'$

Step 5

Solve for $r'$ .

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Step 5.1

Simplify the left side.

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Step 5.1.1

Simplify $- \sin (r) r' - \csc^{2} (x)$ .

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Step 5.1.1.1

Simplify each term.

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Step 5.1.1.1.1

Rewrite $\csc (x)$ in terms of sines and cosines.

$- \sin (r) r' - {(\frac{1}{\sin (x)})}^{2} = e^{r x} r + e^{r x} x r'$

Step 5.1.1.1.2

Apply the product rule to $\frac{1}{\sin (x)}$ .

$- \sin (r) r' - \frac{1^{2}}{\sin^{2} (x)} = e^{r x} r + e^{r x} x r'$

Step 5.1.1.1.3

One to any power is one.

$- \sin (r) r' - \frac{1}{\sin^{2} (x)} = e^{r x} r + e^{r x} x r'$

Step 5.1.1.2

Simplify each term.

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Step 5.1.1.2.1

Rewrite $1$ as $1^{2}$ .

$- \sin (r) r' - \frac{1^{2}}{\sin^{2} (x)} = e^{r x} r + e^{r x} x r'$

Step 5.1.1.2.2

Rewrite $\frac{1^{2}}{\sin^{2} (x)}$ as ${(\frac{1}{\sin (x)})}^{2}$ .

$- \sin (r) r' - {(\frac{1}{\sin (x)})}^{2} = e^{r x} r + e^{r x} x r'$

Step 5.1.1.2.3

Convert from $\frac{1}{\sin (x)}$ to $\csc (x)$ .

$- \sin (r) r' - \csc^{2} (x) = e^{r x} r + e^{r x} x r'$

Step 5.1.1.3

Reorder factors in $- \sin (r) r' - \csc^{2} (x)$ .

$- r' \sin (r) - \csc^{2} (x) = e^{r x} r + e^{r x} x r'$

Step 5.2

Simplify the right side.

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Step 5.2.1

Reorder factors in $e^{r x} r + e^{r x} x r'$ .

$- r' \sin (r) - \csc^{2} (x) = r e^{r x} + x r' e^{r x}$

Step 5.3

Move all terms containing $r'$ to the left side of the equation.

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Step 5.3.1

Subtract $x r' e^{r x}$ from both sides of the equation.

$- r' \sin (r) - \csc^{2} (x) - x r' e^{r x} = r e^{r x}$

Step 5.3.2

Simplify each term.

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Step 5.3.2.1

Rewrite $\csc (x)$ in terms of sines and cosines.

$- r' \sin (r) - {(\frac{1}{\sin (x)})}^{2} - x r' e^{r x} = r e^{r x}$

Step 5.3.2.2

Apply the product rule to $\frac{1}{\sin (x)}$ .

$- r' \sin (r) - \frac{1^{2}}{\sin^{2} (x)} - x r' e^{r x} = r e^{r x}$

Step 5.3.2.3

One to any power is one.

$- r' \sin (r) - \frac{1}{\sin^{2} (x)} - x r' e^{r x} = r e^{r x}$

Step 5.3.3

Simplify each term.

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Step 5.3.3.1

Rewrite $1$ as $1^{2}$ .

$- r' \sin (r) - \frac{1^{2}}{\sin^{2} (x)} - x r' e^{r x} = r e^{r x}$

Step 5.3.3.2

Rewrite $\frac{1^{2}}{\sin^{2} (x)}$ as ${(\frac{1}{\sin (x)})}^{2}$ .

$- r' \sin (r) - {(\frac{1}{\sin (x)})}^{2} - x r' e^{r x} = r e^{r x}$

Step 5.3.3.3

Convert from $\frac{1}{\sin (x)}$ to $\csc (x)$ .

$- r' \sin (r) - \csc^{2} (x) - x r' e^{r x} = r e^{r x}$

Step 5.4

Add $\csc^{2} (x)$ to both sides of the equation.

$- r' \sin (r) - x r' e^{r x} = r e^{r x} + \csc^{2} (x)$

Step 5.5

Factor $r'$ out of $- r' \sin (r) - x r' e^{r x}$ .

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Step 5.5.1

Factor $r'$ out of $- r' \sin (r)$ .

$r' (- 1 \sin (r)) - x r' e^{r x} = r e^{r x} + \csc^{2} (x)$

Step 5.5.2

Factor $r'$ out of $- x r' e^{r x}$ .

$r' (- 1 \sin (r)) + r' (- x e^{r x}) = r e^{r x} + \csc^{2} (x)$

Step 5.5.3

Factor $r'$ out of $r' (- 1 \sin (r)) + r' (- x e^{r x})$ .

$r' (- 1 \sin (r) - x e^{r x}) = r e^{r x} + \csc^{2} (x)$

Step 5.6

Rewrite $- 1 \sin (r)$ as $- \sin (r)$ .

$r' (- \sin (r) - x e^{r x}) = r e^{r x} + \csc^{2} (x)$

Step 5.7

Divide each term in $r' (- \sin (r) - x e^{r x}) = r e^{r x} + \csc^{2} (x)$ by $- \sin (r) - x e^{r x}$ and simplify.

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Step 5.7.1

Divide each term in $r' (- \sin (r) - x e^{r x}) = r e^{r x} + \csc^{2} (x)$ by $- \sin (r) - x e^{r x}$ .

$\frac{r' (- \sin (r) - x e^{r x})}{- \sin (r) - x e^{r x}} = \frac{r e^{r x}}{- \sin (r) - x e^{r x}} + \frac{\csc^{2} (x)}{- \sin (r) - x e^{r x}}$

Step 5.7.2

Simplify the left side.

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Step 5.7.2.1

Cancel the common factor of $- \sin (r) - x e^{r x}$ .

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Step 5.7.2.1.1

Cancel the common factor.

$\frac{r' (- \sin (r) - x e^{r x})}{- \sin (r) - x e^{r x}} = \frac{r e^{r x}}{- \sin (r) - x e^{r x}} + \frac{\csc^{2} (x)}{- \sin (r) - x e^{r x}}$

Step 5.7.2.1.2

Divide $r'$ by $1$ .

$r' = \frac{r e^{r x}}{- \sin (r) - x e^{r x}} + \frac{\csc^{2} (x)}{- \sin (r) - x e^{r x}}$

Step 5.7.3

Simplify the right side.

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Step 5.7.3.1

Simplify each term.

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Step 5.7.3.1.1

Simplify the numerator.

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Step 5.7.3.1.1.1

Rewrite $\csc (x)$ in terms of sines and cosines.

$r' = \frac{r e^{r x}}{- \sin (r) - x e^{r x}} + \frac{{(\frac{1}{\sin (x)})}^{2}}{- \sin (r) - x e^{r x}}$

Step 5.7.3.1.1.2

Apply the product rule to $\frac{1}{\sin (x)}$ .

$r' = \frac{r e^{r x}}{- \sin (r) - x e^{r x}} + \frac{\frac{1^{2}}{\sin^{2} (x)}}{- \sin (r) - x e^{r x}}$

Step 5.7.3.1.1.3

One to any power is one.

$r' = \frac{r e^{r x}}{- \sin (r) - x e^{r x}} + \frac{\frac{1}{\sin^{2} (x)}}{- \sin (r) - x e^{r x}}$

Step 5.7.3.1.2

Multiply the numerator by the reciprocal of the denominator.

$r' = \frac{r e^{r x}}{- \sin (r) - x e^{r x}} + \frac{1}{\sin^{2} (x)} \cdot \frac{1}{- \sin (r) - x e^{r x}}$

Step 5.7.3.1.3

Multiply $\frac{1}{\sin^{2} (x)}$ by $\frac{1}{- \sin (r) - x e^{r x}}$ .

$r' = \frac{r e^{r x}}{- \sin (r) - x e^{r x}} + \frac{1}{\sin^{2} (x) (- \sin (r) - x e^{r x})}$

Step 5.7.3.2

To write $\frac{r e^{r x}}{- \sin (r) - x e^{r x}}$ as a fraction with a common denominator, multiply by $\frac{\sin^{2} (x)}{\sin^{2} (x)}$ .

$r' = \frac{r e^{r x}}{- \sin (r) - x e^{r x}} \cdot \frac{\sin^{2} (x)}{\sin^{2} (x)} + \frac{1}{\sin^{2} (x) (- \sin (r) - x e^{r x})}$

Step 5.7.3.3

Write each expression with a common denominator of $(- \sin (r) - x e^{r x}) \sin^{2} (x)$ , by multiplying each by an appropriate factor of $1$ .

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Step 5.7.3.3.1

Multiply $\frac{r e^{r x}}{- \sin (r) - x e^{r x}}$ by $\frac{\sin^{2} (x)}{\sin^{2} (x)}$ .

$r' = \frac{r e^{r x} \sin^{2} (x)}{(- \sin (r) - x e^{r x}) \sin^{2} (x)} + \frac{1}{\sin^{2} (x) (- \sin (r) - x e^{r x})}$

Step 5.7.3.3.2

Reorder the factors of $(- \sin (r) - x e^{r x}) \sin^{2} (x)$ .

$r' = \frac{r e^{r x} \sin^{2} (x)}{\sin^{2} (x) (- \sin (r) - x e^{r x})} + \frac{1}{\sin^{2} (x) (- \sin (r) - x e^{r x})}$

Step 5.7.3.4

Combine the numerators over the common denominator.

$r' = \frac{r e^{r x} \sin^{2} (x) + 1}{\sin^{2} (x) (- \sin (r) - x e^{r x})}$

Step 5.7.3.5

Factor $- 1$ out of $- \sin (r)$ .

$r' = \frac{r e^{r x} \sin^{2} (x) + 1}{\sin^{2} (x) (- (\sin (r)) - x e^{r x})}$

Step 5.7.3.6

Factor $- 1$ out of $- x e^{r x}$ .

$r' = \frac{r e^{r x} \sin^{2} (x) + 1}{\sin^{2} (x) (- (\sin (r)) - (x e^{r x}))}$

Step 5.7.3.7

Factor $- 1$ out of $- (\sin (r)) - (x e^{r x})$ .

$r' = \frac{r e^{r x} \sin^{2} (x) + 1}{\sin^{2} (x) (- (\sin (r) + x e^{r x}))}$

Step 5.7.3.8

Rewrite negatives.

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Step 5.7.3.8.1

Rewrite $- (\sin (r) + x e^{r x})$ as $- 1 (\sin (r) + x e^{r x})$ .

$r' = \frac{r e^{r x} \sin^{2} (x) + 1}{\sin^{2} (x) (- 1 (\sin (r) + x e^{r x}))}$

Step 5.7.3.8.2

Move the negative in front of the fraction.

$r' = - \frac{r e^{r x} \sin^{2} (x) + 1}{\sin^{2} (x) (\sin (r) + x e^{r x})}$

Step 6

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

$\frac{d r}{d x} = - \frac{r e^{r x} \sin^{2} (x) + 1}{\sin^{2} (x) (\sin (r) + x e^{r x})}$