Calculus Examples

$\sin (4 y) = x$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (\sin (4 y)) = \frac{d}{d x} (x)$

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 $f (x) = \sin (x)$ and $g (x) = 4 y$ .

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Step 2.1.1

To apply the Chain Rule, set $u$ as $4 y$ .

$\frac{d}{d u} [\sin (u)] \frac{d}{d x} [4 y]$

Step 2.1.2

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

$\cos (u) \frac{d}{d x} [4 y]$

Step 2.1.3

Replace all occurrences of $u$ with $4 y$ .

$\cos (4 y) \frac{d}{d x} [4 y]$

Step 2.2

Since $4$ is constant with respect to $x$ , the derivative of $4 y$ with respect to $x$ is $4 \frac{d}{d x} [y]$ .

$\cos (4 y) (4 \frac{d}{d x} [y])$

Step 2.3

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

$\cos (4 y) (4 y')$

Step 2.4

Reorder the factors of $\cos (4 y) (4 y')$ .

$4 \cos (4 y) y'$

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.

$4 \cos (4 y) y' = 1$

Step 5

Divide each term in $4 \cos (4 y) y' = 1$ by $4 \cos (4 y)$ and simplify.

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Step 5.1

Divide each term in $4 \cos (4 y) y' = 1$ by $4 \cos (4 y)$ .

$\frac{4 \cos (4 y) y'}{4 \cos (4 y)} = \frac{1}{4 \cos (4 y)}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $4$ .

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Step 5.2.1.1

Cancel the common factor.

$\frac{4 \cos (4 y) y'}{4 \cos (4 y)} = \frac{1}{4 \cos (4 y)}$

Step 5.2.1.2

Rewrite the expression.

$\frac{\cos (4 y) y'}{\cos (4 y)} = \frac{1}{4 \cos (4 y)}$

Step 5.2.2

Cancel the common factor of $\cos (4 y)$ .

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Step 5.2.2.1

Cancel the common factor.

$\frac{\cos (4 y) y'}{\cos (4 y)} = \frac{1}{4 \cos (4 y)}$

Step 5.2.2.2

Divide $y'$ by $1$ .

$y' = \frac{1}{4 \cos (4 y)}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Separate fractions.

$y' = \frac{1}{4} \cdot \frac{1}{\cos (4 y)}$

Step 5.3.2

Convert from $\frac{1}{\cos (4 y)}$ to $\sec (4 y)$ .

$y' = \frac{1}{4} \sec (4 y)$

Step 5.3.3

Combine $\frac{1}{4}$ and $\sec (4 y)$ .

$y' = \frac{\sec (4 y)}{4}$

Step 6

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

$\frac{d y}{d x} = \frac{\sec (4 y)}{4}$