Calculus Examples

$y = \frac{2 u}{1 - 4 u}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d u} (y) = \frac{d}{d u} (\frac{2 u}{1 - 4 u})$

Step 2

The derivative of $y$ with respect to $u$ is $y'$ .

$y'$

Step 3

Differentiate the right side of the equation.

Tap for more steps...

Step 3.1

Since $2$ is constant with respect to $u$ , the derivative of $\frac{2 u}{1 - 4 u}$ with respect to $u$ is $2 \frac{d}{d u} [\frac{u}{1 - 4 u}]$ .

$2 \frac{d}{d u} [\frac{u}{1 - 4 u}]$

Step 3.2

Differentiate using the Quotient Rule which states that $\frac{d}{d u} [\frac{f (u)}{g (u)}]$ is $\frac{g (u) \frac{d}{d u} [f (u)] - f (u) \frac{d}{d u} [g (u)]}{{g (u)}^{2}}$ where $f (u) = u$ and $g (u) = 1 - 4 u$ .

$2 \frac{(1 - 4 u) \frac{d}{d u} [u] - u \frac{d}{d u} [1 - 4 u]}{{(1 - 4 u)}^{2}}$

Step 3.3

Differentiate.

Tap for more steps...

Step 3.3.1

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

$2 \frac{(1 - 4 u) \cdot 1 - u \frac{d}{d u} [1 - 4 u]}{{(1 - 4 u)}^{2}}$

Step 3.3.2

Multiply $1 - 4 u$ by $1$ .

$2 \frac{1 - 4 u - u \frac{d}{d u} [1 - 4 u]}{{(1 - 4 u)}^{2}}$

Step 3.3.3

By the Sum Rule, the derivative of $1 - 4 u$ with respect to $u$ is $\frac{d}{d u} [1] + \frac{d}{d u} [- 4 u]$ .

$2 \frac{1 - 4 u - u (\frac{d}{d u} [1] + \frac{d}{d u} [- 4 u])}{{(1 - 4 u)}^{2}}$

Step 3.3.4

Since $1$ is constant with respect to $u$ , the derivative of $1$ with respect to $u$ is $0$ .

$2 \frac{1 - 4 u - u (0 + \frac{d}{d u} [- 4 u])}{{(1 - 4 u)}^{2}}$

Step 3.3.5

Add $0$ and $\frac{d}{d u} [- 4 u]$ .

$2 \frac{1 - 4 u - u \frac{d}{d u} [- 4 u]}{{(1 - 4 u)}^{2}}$

Step 3.3.6

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

$2 \frac{1 - 4 u - u (- 4 \frac{d}{d u} [u])}{{(1 - 4 u)}^{2}}$

Step 3.3.7

Multiply $- 4$ by $- 1$ .

$2 \frac{1 - 4 u + 4 u \frac{d}{d u} [u]}{{(1 - 4 u)}^{2}}$

Step 3.3.8

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

$2 \frac{1 - 4 u + 4 u \cdot 1}{{(1 - 4 u)}^{2}}$

Step 3.3.9

Simplify terms.

Tap for more steps...

Step 3.3.9.1

Multiply $4$ by $1$ .

$2 \frac{1 - 4 u + 4 u}{{(1 - 4 u)}^{2}}$

Step 3.3.9.2

Add $- 4 u$ and $4 u$ .

$2 \frac{1 + 0}{{(1 - 4 u)}^{2}}$

Step 3.3.9.3

Add $1$ and $0$ .

$2 \frac{1}{{(1 - 4 u)}^{2}}$

Step 3.3.9.4

Combine $2$ and $\frac{1}{{(1 - 4 u)}^{2}}$ .

$\frac{2}{{(1 - 4 u)}^{2}}$

Step 3.3.9.5

Reorder terms.

$\frac{2}{{(- 4 u + 1)}^{2}}$

Step 4

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

$y' = \frac{2}{{(- 4 u + 1)}^{2}}$

Step 5

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

$\frac{d y}{d u} = \frac{2}{{(- 4 u + 1)}^{2}}$