Calculus Examples

cos (2 y) = x

$\cos (2 y) = x$

Step 1

Differentiate both sides of the equation.

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

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

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))]

$\frac{d}{d x} [f (g (x))]$ 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

$\frac{d y}{d x}$ .

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