Calculus Examples

$f (x) = x^{2} + {(200 - x)}^{2}$

Step 1

Rewrite ${(200 - x)}^{2}$ as $(200 - x) (200 - x)$ .

$\frac{d}{d x} [x^{2} + (200 - x) (200 - x)]$

Step 2

Expand $(200 - x) (200 - x)$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

$\frac{d}{d x} [x^{2} + 200 (200 - x) - x (200 - x)]$

Step 2.2

Apply the distributive property.

$\frac{d}{d x} [x^{2} + 200 \cdot 200 + 200 (- x) - x (200 - x)]$

Step 2.3

Apply the distributive property.

$\frac{d}{d x} [x^{2} + 200 \cdot 200 + 200 (- x) - x \cdot 200 - x (- x)]$

Step 3

Simplify and combine like terms.

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Step 3.1

Simplify each term.

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Step 3.1.1

Multiply $200$ by $200$ .

$\frac{d}{d x} [x^{2} + 40000 + 200 (- x) - x \cdot 200 - x (- x)]$

Step 3.1.2

Multiply $- 1$ by $200$ .

$\frac{d}{d x} [x^{2} + 40000 - 200 x - x \cdot 200 - x (- x)]$

Step 3.1.3

Multiply $200$ by $- 1$ .

$\frac{d}{d x} [x^{2} + 40000 - 200 x - 200 x - x (- x)]$

Step 3.1.4

Rewrite using the commutative property of multiplication.

$\frac{d}{d x} [x^{2} + 40000 - 200 x - 200 x - 1 \cdot - 1 x \cdot x]$

Step 3.1.5

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

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Step 3.1.5.1

Move $x$ .

$\frac{d}{d x} [x^{2} + 40000 - 200 x - 200 x - 1 \cdot - 1 (x \cdot x)]$

Step 3.1.5.2

Multiply $x$ by $x$ .

$\frac{d}{d x} [x^{2} + 40000 - 200 x - 200 x - 1 \cdot - 1 x^{2}]$

Step 3.1.6

Multiply $- 1$ by $- 1$ .

$\frac{d}{d x} [x^{2} + 40000 - 200 x - 200 x + 1 x^{2}]$

Step 3.1.7

Multiply $x^{2}$ by $1$ .

$\frac{d}{d x} [x^{2} + 40000 - 200 x - 200 x + x^{2}]$

Step 3.2

Subtract $200 x$ from $- 200 x$ .

$\frac{d}{d x} [x^{2} + 40000 - 400 x + x^{2}]$

Step 4

Differentiate.

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Step 4.1

By the Sum Rule, the derivative of $x^{2} + 40000 - 400 x + x^{2}$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [40000] + \frac{d}{d x} [- 400 x] + \frac{d}{d x} [x^{2}]$ .

$\frac{d}{d x} [x^{2}] + \frac{d}{d x} [40000] + \frac{d}{d x} [- 400 x] + \frac{d}{d x} [x^{2}]$

Step 4.2

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

$2 x + \frac{d}{d x} [40000] + \frac{d}{d x} [- 400 x] + \frac{d}{d x} [x^{2}]$

Step 4.3

Since $40000$ is constant with respect to $x$ , the derivative of $40000$ with respect to $x$ is $0$ .

$2 x + 0 + \frac{d}{d x} [- 400 x] + \frac{d}{d x} [x^{2}]$

Step 5

Evaluate $\frac{d}{d x} [- 400 x]$ .

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Step 5.1

Since $- 400$ is constant with respect to $x$ , the derivative of $- 400 x$ with respect to $x$ is $- 400 \frac{d}{d x} [x]$ .

$2 x + 0 - 400 \frac{d}{d x} [x] + \frac{d}{d x} [x^{2}]$

Step 5.2

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

$2 x + 0 - 400 \cdot 1 + \frac{d}{d x} [x^{2}]$

Step 5.3

Multiply $- 400$ by $1$ .

$2 x + 0 - 400 + \frac{d}{d x} [x^{2}]$

Step 6

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

$2 x + 0 - 400 + 2 x$

Step 7

Combine terms.

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Step 7.1

Add $2 x$ and $0$ .

$2 x - 400 + 2 x$

Step 7.2

Add $2 x$ and $2 x$ .

$4 x - 400$