Calculus Examples

$\cos (2 y) = x$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (\cos (2 y)) = \frac{d}{d x} (x)$

Step 2

Differentiate the left side of the equation.

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Step 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 $g (x) = 2 y$ .

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Step 2.1.1

To apply the Chain Rule, set $u$ as $2 y$ .

$\frac{d}{d u} [\cos (u)] \frac{d}{d x} [2 y]$

Step 2.1.2

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

$- \sin (u) \frac{d}{d x} [2 y]$

Step 2.1.3

Replace all occurrences of $u$ with $2 y$ .

$- \sin (2 y) \frac{d}{d x} [2 y]$

Step 2.2

Differentiate using the Constant Multiple Rule.

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Step 2.2.1

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

$- \sin (2 y) (2 \frac{d}{d x} [y])$

Step 2.2.2

Multiply $2$ by $- 1$ .

$- 2 \sin (2 y) \frac{d}{d x} [y]$

Step 2.3

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

$- 2 \sin (2 y) y'$

Step 3

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

$1$

Step 4

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

$- 2 \sin (2 y) y' = 1$

Step 5

Divide each term in $- 2 \sin (2 y) y' = 1$ by $- 2 \sin (2 y)$ and simplify.

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Step 5.1

Divide each term in $- 2 \sin (2 y) y' = 1$ by $- 2 \sin (2 y)$ .

$\frac{- 2 \sin (2 y) y'}{- 2 \sin (2 y)} = \frac{1}{- 2 \sin (2 y)}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $- 2$ .

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Step 5.2.1.1

Cancel the common factor.

$\frac{- 2 \sin (2 y) y'}{- 2 \sin (2 y)} = \frac{1}{- 2 \sin (2 y)}$

Step 5.2.1.2

Rewrite the expression.

$\frac{\sin (2 y) y'}{\sin (2 y)} = \frac{1}{- 2 \sin (2 y)}$

Step 5.2.2

Cancel the common factor of $\sin (2 y)$ .

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Step 5.2.2.1

Cancel the common factor.

$\frac{\sin (2 y) y'}{\sin (2 y)} = \frac{1}{- 2 \sin (2 y)}$

Step 5.2.2.2

Divide $y'$ by $1$ .

$y' = \frac{1}{- 2 \sin (2 y)}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Separate fractions.

$y' = \frac{1}{- 2} \cdot \frac{1}{\sin (2 y)}$

Step 5.3.2

Convert from $\frac{1}{\sin (2 y)}$ to $\csc (2 y)$ .

$y' = \frac{1}{- 2} \csc (2 y)$

Step 5.3.3

Move the negative in front of the fraction.

$y' = - \frac{1}{2} \csc (2 y)$

Step 5.3.4

Combine $\csc (2 y)$ and $\frac{1}{2}$ .

$y' = - \frac{\csc (2 y)}{2}$

Step 6

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

$\frac{d y}{d x} = - \frac{\csc (2 y)}{2}$