Calculus Examples

$y = {(4 x - 1)}^{2}$

Step 1

Differentiate both sides of the equation.

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

Step 2

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

$y'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

Rewrite ${(4 x - 1)}^{2}$ as $(4 x - 1) (4 x - 1)$ .

$\frac{d}{d x} [(4 x - 1) (4 x - 1)]$

Step 3.2

Expand $(4 x - 1) (4 x - 1)$ using the FOIL Method.

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Step 3.2.1

Apply the distributive property.

$\frac{d}{d x} [4 x (4 x - 1) - 1 (4 x - 1)]$

Step 3.2.2

Apply the distributive property.

$\frac{d}{d x} [4 x (4 x) + 4 x \cdot - 1 - 1 (4 x - 1)]$

Step 3.2.3

Apply the distributive property.

$\frac{d}{d x} [4 x (4 x) + 4 x \cdot - 1 - 1 (4 x) - 1 \cdot - 1]$

Step 3.3

Simplify and combine like terms.

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Step 3.3.1

Simplify each term.

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Step 3.3.1.1

Rewrite using the commutative property of multiplication.

$\frac{d}{d x} [4 \cdot 4 x \cdot x + 4 x \cdot - 1 - 1 (4 x) - 1 \cdot - 1]$

Step 3.3.1.2

Multiply $x$ by $x$ by adding the exponents.

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Step 3.3.1.2.1

Move $x$ .

$\frac{d}{d x} [4 \cdot 4 (x \cdot x) + 4 x \cdot - 1 - 1 (4 x) - 1 \cdot - 1]$

Step 3.3.1.2.2

Multiply $x$ by $x$ .

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

Step 3.3.1.3

Multiply $4$ by $4$ .

$\frac{d}{d x} [16 x^{2} + 4 x \cdot - 1 - 1 (4 x) - 1 \cdot - 1]$

Step 3.3.1.4

Multiply $- 1$ by $4$ .

$\frac{d}{d x} [16 x^{2} - 4 x - 1 (4 x) - 1 \cdot - 1]$

Step 3.3.1.5

Multiply $4$ by $- 1$ .

$\frac{d}{d x} [16 x^{2} - 4 x - 4 x - 1 \cdot - 1]$

Step 3.3.1.6

Multiply $- 1$ by $- 1$ .

$\frac{d}{d x} [16 x^{2} - 4 x - 4 x + 1]$

Step 3.3.2

Subtract $4 x$ from $- 4 x$ .

$\frac{d}{d x} [16 x^{2} - 8 x + 1]$

Step 3.4

By the Sum Rule, the derivative of $16 x^{2} - 8 x + 1$ with respect to $x$ is $\frac{d}{d x} [16 x^{2}] + \frac{d}{d x} [- 8 x] + \frac{d}{d x} [1]$ .

$\frac{d}{d x} [16 x^{2}] + \frac{d}{d x} [- 8 x] + \frac{d}{d x} [1]$

Step 3.5

Since $16$ is constant with respect to $x$ , the derivative of $16 x^{2}$ with respect to $x$ is $16 \frac{d}{d x} [x^{2}]$ .

$16 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 8 x] + \frac{d}{d x} [1]$

Step 3.6

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

$16 (2 x) + \frac{d}{d x} [- 8 x] + \frac{d}{d x} [1]$

Step 3.7

Multiply $2$ by $16$ .

$32 x + \frac{d}{d x} [- 8 x] + \frac{d}{d x} [1]$

Step 3.8

Since $- 8$ is constant with respect to $x$ , the derivative of $- 8 x$ with respect to $x$ is $- 8 \frac{d}{d x} [x]$ .

$32 x - 8 \frac{d}{d x} [x] + \frac{d}{d x} [1]$

Step 3.9

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

$32 x - 8 \cdot 1 + \frac{d}{d x} [1]$

Step 3.10

Multiply $- 8$ by $1$ .

$32 x - 8 + \frac{d}{d x} [1]$

Step 3.11

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

$32 x - 8 + 0$

Step 3.12

Add $32 x - 8$ and $0$ .

$32 x - 8$

Step 4

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

$y' = 32 x - 8$

Step 5

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

$\frac{d y}{d x} = 32 x - 8$