Calculus Examples

$x = \csc (y)$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (x) = \frac{d}{d x} (\csc (y))$

Step 2

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

$1$

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 $f (x) = \csc (x)$ and $g (x) = y$ .

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Step 3.1.1

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

$\frac{d}{d u} [\csc (u)] \frac{d}{d x} [y]$

Step 3.1.2

The derivative of $\csc (u)$ with respect to $u$ is $- \csc (u) \cot (u)$ .

$- \csc (u) \cot (u) \frac{d}{d x} [y]$

Step 3.1.3

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

$- \csc (y) \cot (y) \frac{d}{d x} [y]$

Step 3.2

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

$- \csc (y) \cot (y) y'$

Step 3.3

Reorder the factors of $- \csc (y) \cot (y) y'$ .

$- \cot (y) \csc (y) y'$

Step 4

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

$1 = - \cot (y) \csc (y) y'$

Step 5

Solve for $y'$ .

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Step 5.1

Rewrite the equation as $- \cot (y) \csc (y) y' = 1$ .

$- \cot (y) \csc (y) y' = 1$

Step 5.2

Divide each term in $- \cot (y) \csc (y) y' = 1$ by $- \cot (y) \csc (y)$ and simplify.

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Step 5.2.1

Divide each term in $- \cot (y) \csc (y) y' = 1$ by $- \cot (y) \csc (y)$ .

$\frac{- \cot (y) \csc (y) y'}{- \cot (y) \csc (y)} = \frac{1}{- \cot (y) \csc (y)}$

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Dividing two negative values results in a positive value.

$\frac{\cot (y) \csc (y) y'}{\cot (y) \csc (y)} = \frac{1}{- \cot (y) \csc (y)}$

Step 5.2.2.2

Cancel the common factor of $\cot (y)$ .

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Step 5.2.2.2.1

Cancel the common factor.

$\frac{\cot (y) \csc (y) y'}{\cot (y) \csc (y)} = \frac{1}{- \cot (y) \csc (y)}$

Step 5.2.2.2.2

Rewrite the expression.

$\frac{\csc (y) y'}{\csc (y)} = \frac{1}{- \cot (y) \csc (y)}$

Step 5.2.2.3

Cancel the common factor of $\csc (y)$ .

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Step 5.2.2.3.1

Cancel the common factor.

$\frac{\csc (y) y'}{\csc (y)} = \frac{1}{- \cot (y) \csc (y)}$

Step 5.2.2.3.2

Divide $y'$ by $1$ .

$y' = \frac{1}{- \cot (y) \csc (y)}$

Step 5.2.3

Simplify the right side.

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Step 5.2.3.1

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

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Step 5.2.3.1.1

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

$y' = \frac{- 1 (- 1)}{- \cot (y) \csc (y)}$

Step 5.2.3.1.2

Move the negative in front of the fraction.

$y' = - \frac{1}{\cot (y) \csc (y)}$

Step 5.2.3.2

Separate fractions.

$y' = - (\frac{1}{\csc (y)} \cdot \frac{1}{\cot (y)})$

Step 5.2.3.3

Rewrite $\cot (y)$ in terms of sines and cosines.

$y' = - (\frac{1}{\csc (y)} \cdot \frac{1}{\frac{\cos (y)}{\sin (y)}})$

Step 5.2.3.4

Multiply by the reciprocal of the fraction to divide by $\frac{\cos (y)}{\sin (y)}$ .

$y' = - (\frac{1}{\csc (y)} \cdot \frac{\sin (y)}{\cos (y)})$

Step 5.2.3.5

Convert from $\frac{\sin (y)}{\cos (y)}$ to $\tan (y)$ .

$y' = - (\frac{1}{\csc (y)} \tan (y))$

Step 5.2.3.6

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

$y' = - (\frac{1}{\frac{1}{\sin (y)}} \tan (y))$

Step 5.2.3.7

Multiply by the reciprocal of the fraction to divide by $\frac{1}{\sin (y)}$ .

$y' = - (1 \sin (y) \tan (y))$

Step 5.2.3.8

Multiply $\sin (y)$ by $1$ .

$y' = - \sin (y) \tan (y)$

Step 6

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

$\frac{d y}{d x} = - \sin (y) \tan (y)$