Algebra Examples

$\sin (2 x) = 4 x + 3 y$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (\sin (2 x)) = \frac{d}{d x} (4 x + 3 y)$

Step 2

Differentiate the left side of the equation.

Tap for more steps...

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 $g (x) = 2 x$ .

Tap for more steps...

Step 2.1.1

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

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

Step 2.1.2

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

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

Step 2.1.3

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

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

Step 2.2

Differentiate.

Tap for more steps...

Step 2.2.1

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

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

Step 2.2.2

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

$\cos (2 x) (2 \cdot 1)$

Step 2.2.3

Simplify the expression.

Tap for more steps...

Step 2.2.3.1

Multiply $2$ by $1$ .

$\cos (2 x) \cdot 2$

Step 2.2.3.2

Move $2$ to the left of $\cos (2 x)$ .

$2 \cos (2 x)$

Step 3

Differentiate the right side of the equation.

Tap for more steps...

Step 3.1

By the Sum Rule, the derivative of $4 x + 3 y$ with respect to $x$ is $\frac{d}{d x} [4 x] + \frac{d}{d x} [3 y]$ .

$\frac{d}{d x} [4 x] + \frac{d}{d x} [3 y]$

Step 3.2

Evaluate $\frac{d}{d x} [4 x]$ .

Tap for more steps...

Step 3.2.1

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

$4 \frac{d}{d x} [x] + \frac{d}{d x} [3 y]$

Step 3.2.2

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

$4 \cdot 1 + \frac{d}{d x} [3 y]$

Step 3.2.3

Multiply $4$ by $1$ .

$4 + \frac{d}{d x} [3 y]$

Step 3.3

Evaluate $\frac{d}{d x} [3 y]$ .

Tap for more steps...

Step 3.3.1

Since $3$ is constant with respect to $x$ , the derivative of $3 y$ with respect to $x$ is $3 \frac{d}{d x} [y]$ .

$4 + 3 \frac{d}{d x} [y]$

Step 3.3.2

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

$4 + 3 y'$

Step 3.4

Reorder terms.

$3 y' + 4$

Step 4

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

$2 \cos (2 x) = 3 y' + 4$

Step 5

Solve for $y'$ .

Tap for more steps...

Step 5.1

Rewrite the equation as $3 y' + 4 = 2 \cos (2 x)$ .

$3 y' + 4 = 2 \cos (2 x)$

Step 5.2

Use the double-angle identity to transform $\cos (2 x)$ to $1 - 2 \sin^{2} (x)$ .

$3 y' + 4 = 2 (1 - 2 \sin^{2} (x))$

Step 5.3

Subtract $4$ from both sides of the equation.

$3 y' = 2 (1 - 2 \sin^{2} (x)) - 4$

Step 5.4

Simplify the right side.

Tap for more steps...

Step 5.4.1

Simplify $2 (1 - 2 \sin^{2} (x)) - 4$ .

Tap for more steps...

Step 5.4.1.1

Simplify each term.

Tap for more steps...

Step 5.4.1.1.1

Apply the distributive property.

$3 y' = 2 \cdot 1 + 2 (- 2 \sin^{2} (x)) - 4$

Step 5.4.1.1.2

Multiply $2$ by $1$ .

$3 y' = 2 + 2 (- 2 \sin^{2} (x)) - 4$

Step 5.4.1.1.3

Multiply $- 2$ by $2$ .

$3 y' = 2 - 4 \sin^{2} (x) - 4$

Step 5.4.1.2

Subtract $4$ from $2$ .

$3 y' = - 2 - 4 \sin^{2} (x)$

Step 5.5

Divide each term in $3 y' = - 2 - 4 \sin^{2} (x)$ by $3$ and simplify.

Tap for more steps...

Step 5.5.1

Divide each term in $3 y' = - 2 - 4 \sin^{2} (x)$ by $3$ .

$\frac{3 y'}{3} = \frac{- 2}{3} + \frac{- 4 \sin^{2} (x)}{3}$

Step 5.5.2

Simplify the left side.

Tap for more steps...

Step 5.5.2.1

Cancel the common factor of $3$ .

Tap for more steps...

Step 5.5.2.1.1

Cancel the common factor.

$\frac{3 y'}{3} = \frac{- 2}{3} + \frac{- 4 \sin^{2} (x)}{3}$

Step 5.5.2.1.2

Divide $y'$ by $1$ .

$y' = \frac{- 2}{3} + \frac{- 4 \sin^{2} (x)}{3}$

Step 5.5.3

Simplify the right side.

Tap for more steps...

Step 5.5.3.1

Simplify each term.

Tap for more steps...

Step 5.5.3.1.1

Move the negative in front of the fraction.

$y' = - \frac{2}{3} + \frac{- 4 \sin^{2} (x)}{3}$

Step 5.5.3.1.2

Move the negative in front of the fraction.

$y' = - \frac{2}{3} - \frac{4 \sin^{2} (x)}{3}$

Step 6

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

$\frac{d y}{d x} = - \frac{2}{3} - \frac{4 \sin^{2} (x)}{3}$