Algebra Examples

$y = \cos^{2} (x)$

Step 1

Differentiate both sides of the equation.

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

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.

Tap for more steps...

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 $f (y) = y^{2}$ and $g (y) = \cos (x)$ .

Tap for more steps...

Step 3.1.1

To apply the Chain Rule, set $u_{1}$ as $\cos (x)$ .

$\frac{d}{d u_{1}} [{u_{1}}^{2}] \frac{d}{d y} [\cos (x)]$

Step 3.1.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{1}} [{u_{1}}^{n}]$ is $n {u_{1}}^{n - 1}$ where $n = 2$ .

$2 u_{1} \frac{d}{d y} [\cos (x)]$

Step 3.1.3

Replace all occurrences of $u_{1}$ with $\cos (x)$ .

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

Step 3.2

Differentiate using the chain rule, which states that $\frac{d}{d y} [f (g (y))]$ is $f' (g (y)) g' (y)$ where $f (y) = \cos (y)$ and $g (y) = x$ .

Tap for more steps...

Step 3.2.1

To apply the Chain Rule, set $u_{2}$ as $x$ .

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

Step 3.2.2

The derivative of $\cos (u_{2})$ with respect to $u_{2}$ is $- \sin (u_{2})$ .

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

Step 3.2.3

Replace all occurrences of $u_{2}$ with $x$ .

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

Step 3.3

Multiply $- 1$ by $2$ .

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

Step 3.4

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

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

Step 4

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

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

Step 5

Solve for $x'$ .

Tap for more steps...

Step 5.1

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

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

Step 5.2

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

Tap for more steps...

Step 5.2.1

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

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

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.

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

Step 5.2.2.1.2

Rewrite the expression.

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

Step 5.2.2.2

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

Tap for more steps...

Step 5.2.2.2.1

Cancel the common factor.

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

Step 5.2.2.2.2

Rewrite the expression.

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

Step 5.2.2.3

Cancel the common factor of $\sin (x)$ .

Tap for more steps...

Step 5.2.2.3.1

Cancel the common factor.

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

Step 5.2.2.3.2

Divide $x'$ by $1$ .

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

Step 5.2.3

Simplify the right side.

Tap for more steps...

Step 5.2.3.1

Separate fractions.

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

Step 5.2.3.2

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

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

Step 5.2.3.3

Separate fractions.

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

Step 5.2.3.4

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

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

Step 5.2.3.5

Simplify the expression.

Tap for more steps...

Step 5.2.3.5.1

Move the negative in front of the fraction.

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

Step 5.2.3.5.2

Add parentheses.

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

Step 5.2.3.5.3

Add parentheses.

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

Step 5.2.3.6

Combine to $c s c (2 x)$ .

$x' = - \csc (2 x)$

Step 6

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

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