Calculus Examples

$x = \sec (2 y)$

Step 1

Differentiate both sides of the equation.

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

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.

Tap for more steps...

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 $f (x) = \sec (x)$ and $g (x) = 2 y$ .

Tap for more steps...

Step 3.1.1

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

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

Step 3.1.2

The derivative of $\sec (u)$ with respect to $u$ is $\sec (u) \tan (u)$ .

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

Step 3.1.3

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

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

Step 3.2

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

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

Step 3.3

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

$\sec (2 y) \tan (2 y) (2 y')$

Step 3.4

Reorder the factors of $\sec (2 y) \tan (2 y) (2 y')$ .

$2 \sec (2 y) \tan (2 y) y'$

Step 4

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

$1 = 2 \sec (2 y) \tan (2 y) y'$

Step 5

Solve for $y'$ .

Tap for more steps...

Step 5.1

Rewrite the equation as $2 \sec (2 y) \tan (2 y) y' = 1$ .

$2 \sec (2 y) \tan (2 y) y' = 1$

Step 5.2

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

Tap for more steps...

Step 5.2.1

Divide each term in $2 \sec (2 y) \tan (2 y) y' = 1$ by $2 \sec (2 y) \tan (2 y)$ .

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

Step 5.2.2

Simplify the left side.

Tap for more steps...

Step 5.2.2.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 5.2.2.1.1

Cancel the common factor.

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

Step 5.2.2.1.2

Rewrite the expression.

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

Step 5.2.2.2

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

Tap for more steps...

Step 5.2.2.2.1

Cancel the common factor.

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

Step 5.2.2.2.2

Rewrite the expression.

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

Step 5.2.2.3

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

Tap for more steps...

Step 5.2.2.3.1

Cancel the common factor.

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

Step 5.2.2.3.2

Divide $y'$ by $1$ .

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

Step 5.2.3

Simplify the right side.

Tap for more steps...

Step 5.2.3.1

Separate fractions.

$y' = \frac{1}{2 \tan (2 y)} \cdot \frac{1}{\sec (2 y)}$

Step 5.2.3.2

Rewrite $\sec (2 y)$ in terms of sines and cosines.

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

Step 5.2.3.3

Multiply by the reciprocal of the fraction to divide by $\frac{1}{\cos (2 y)}$ .

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

Step 5.2.3.4

Multiply $\cos (2 y)$ by $1$ .

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

Step 5.2.3.5

Separate fractions.

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

Step 5.2.3.6

Rewrite $\tan (2 y)$ in terms of sines and cosines.

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

Step 5.2.3.7

Multiply by the reciprocal of the fraction to divide by $\frac{\sin (2 y)}{\cos (2 y)}$ .

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

Step 5.2.3.8

Convert from $\frac{\cos (2 y)}{\sin (2 y)}$ to $\cot (2 y)$ .

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

Step 5.2.3.9

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

$y' = \frac{\cot (2 y)}{2} \cos (2 y)$

Step 5.2.3.10

Combine $\frac{\cot (2 y)}{2}$ and $\cos (2 y)$ .

$y' = \frac{\cot (2 y) \cos (2 y)}{2}$

Step 6

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

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