Calculus Examples

$y = \sin (2 x)$

Step 1

Differentiate both sides of the equation.

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

Step 2

Differentiate using the Power Rule which states that $\frac{d}{d y} [y^{n}]$ is $n y^{n - 1}$ 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 $\frac{d}{d y} [f (g (y))]$ is $f' (g (y)) g' (y)$ where $f (y) = \sin (y)$ and $g (y) = 2 x$ .

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Step 3.1.1

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

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

Step 3.1.2

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

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

Step 3.1.3

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

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

Step 3.2

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

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

Step 3.3

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

$\cos (2 x) (2 x')$

Step 3.4

Reorder the factors of $\cos (2 x) (2 x')$ .

$2 \cos (2 x) x'$

Step 4

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

$1 = 2 \cos (2 x) x'$

Step 5

Solve for $x'$ .

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Step 5.1

Rewrite the equation as $2 \cos (2 x) x' = 1$ .

$2 \cos (2 x) x' = 1$

Step 5.2

Divide each term in $2 \cos (2 x) x' = 1$ by $2 \cos (2 x)$ and simplify.

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Step 5.2.1

Divide each term in $2 \cos (2 x) x' = 1$ by $2 \cos (2 x)$ .

$\frac{2 \cos (2 x) x'}{2 \cos (2 x)} = \frac{1}{2 \cos (2 x)}$

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Cancel the common factor of $2$ .

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Step 5.2.2.1.1

Cancel the common factor.

$\frac{2 \cos (2 x) x'}{2 \cos (2 x)} = \frac{1}{2 \cos (2 x)}$

Step 5.2.2.1.2

Rewrite the expression.

$\frac{\cos (2 x) x'}{\cos (2 x)} = \frac{1}{2 \cos (2 x)}$

Step 5.2.2.2

Cancel the common factor of $\cos (2 x)$ .

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Step 5.2.2.2.1

Cancel the common factor.

$\frac{\cos (2 x) x'}{\cos (2 x)} = \frac{1}{2 \cos (2 x)}$

Step 5.2.2.2.2

Divide $x'$ by $1$ .

$x' = \frac{1}{2 \cos (2 x)}$

Step 5.2.3

Simplify the right side.

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Step 5.2.3.1

Separate fractions.

$x' = \frac{1}{2} \cdot \frac{1}{\cos (2 x)}$

Step 5.2.3.2

Convert from $\frac{1}{\cos (2 x)}$ to $\sec (2 x)$ .

$x' = \frac{1}{2} \sec (2 x)$

Step 5.2.3.3

Combine $\frac{1}{2}$ and $\sec (2 x)$ .

$x' = \frac{\sec (2 x)}{2}$

Step 6

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

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