Calculus Examples

$C = 0.001 x^{3} + 5 x + 250$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d C} (C) = \frac{d}{d C} (0.001 x^{3} + 5 x + 250)$

Step 2

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

$1$

Step 3

Differentiate the right side of the equation.

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Step 3.1

By the Sum Rule, the derivative of $0.001 x^{3} + 5 x + 250$ with respect to $C$ is $\frac{d}{d C} [0.001 x^{3}] + \frac{d}{d C} [5 x] + \frac{d}{d C} [250]$ .

$\frac{d}{d C} [0.001 x^{3}] + \frac{d}{d C} [5 x] + \frac{d}{d C} [250]$

Step 3.2

Evaluate $\frac{d}{d C} [0.001 x^{3}]$ .

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Step 3.2.1

Since $0.001$ is constant with respect to $C$ , the derivative of $0.001 x^{3}$ with respect to $C$ is $0.001 \frac{d}{d C} [x^{3}]$ .

$0.001 \frac{d}{d C} [x^{3}] + \frac{d}{d C} [5 x] + \frac{d}{d C} [250]$

Step 3.2.2

Differentiate using the chain rule, which states that $\frac{d}{d C} [f (g (C))]$ is $f' (g (C)) g' (C)$ where $f (C) = C^{3}$ and $g (C) = x$ .

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Step 3.2.2.1

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

$0.001 (\frac{d}{d u} [u^{3}] \frac{d}{d C} [x]) + \frac{d}{d C} [5 x] + \frac{d}{d C} [250]$

Step 3.2.2.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 3$ .

$0.001 (3 u^{2} \frac{d}{d C} [x]) + \frac{d}{d C} [5 x] + \frac{d}{d C} [250]$

Step 3.2.2.3

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

$0.001 (3 x^{2} \frac{d}{d C} [x]) + \frac{d}{d C} [5 x] + \frac{d}{d C} [250]$

Step 3.2.3

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

$0.001 (3 x^{2} x') + \frac{d}{d C} [5 x] + \frac{d}{d C} [250]$

Step 3.2.4

Multiply $3$ by $0.001$ .

$0.003 x^{2} x' + \frac{d}{d C} [5 x] + \frac{d}{d C} [250]$

Step 3.3

Evaluate $\frac{d}{d C} [5 x]$ .

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Step 3.3.1

Since $5$ is constant with respect to $C$ , the derivative of $5 x$ with respect to $C$ is $5 \frac{d}{d C} [x]$ .

$0.003 x^{2} x' + 5 \frac{d}{d C} [x] + \frac{d}{d C} [250]$

Step 3.3.2

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

$0.003 x^{2} x' + 5 x' + \frac{d}{d C} [250]$

Step 3.4

Differentiate using the Constant Rule.

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Step 3.4.1

Since $250$ is constant with respect to $C$ , the derivative of $250$ with respect to $C$ is $0$ .

$0.003 x^{2} x' + 5 x' + 0$

Step 3.4.2

Add $0.003 x^{2} x' + 5 x'$ and $0$ .

$0.003 x^{2} x' + 5 x'$

Step 4

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

$1 = 0.003 x^{2} x' + 5 x'$

Step 5

Solve for $x'$ .

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Step 5.1

Rewrite the equation as $0.003 x^{2} x' + 5 x' = 1$ .

$0.003 x^{2} x' + 5 x' = 1$

Step 5.2

Factor $0.001 x'$ out of $0.003 x^{2} x' + 5 x'$ .

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Step 5.2.1

Move $x^{2}$ .

$0.003 x' x^{2} + 5 x' = 1$

Step 5.2.2

Factor $0.001 x'$ out of $0.003 x' x^{2}$ .

$0.001 x' (3 x^{2}) + 5 x' = 1$

Step 5.2.3

Factor $0.001 x'$ out of $5 x'$ .

$0.001 x' (3 x^{2}) + 0.001 x' \cdot 5000 = 1$

Step 5.2.4

Factor $0.001 x'$ out of $0.001 x' (3 x^{2}) + 0.001 x' \cdot 5000$ .

$0.001 x' (3 x^{2} + 5000) = 1$

Step 5.3

Divide each term in $0.001 x' (3 x^{2} + 5000) = 1$ by $0.001 (3 x^{2} + 5000)$ and simplify.

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Step 5.3.1

Divide each term in $0.001 x' (3 x^{2} + 5000) = 1$ by $0.001 (3 x^{2} + 5000)$ .

$\frac{0.001 x' (3 x^{2} + 5000)}{0.001 (3 x^{2} + 5000)} = \frac{1}{0.001 (3 x^{2} + 5000)}$

Step 5.3.2

Simplify the left side.

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Step 5.3.2.1

Cancel the common factor of $0.001$ .

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Step 5.3.2.1.1

Cancel the common factor.

$\frac{0.001 x' (3 x^{2} + 5000)}{0.001 (3 x^{2} + 5000)} = \frac{1}{0.001 (3 x^{2} + 5000)}$

Step 5.3.2.1.2

Rewrite the expression.

$\frac{x' (3 x^{2} + 5000)}{3 x^{2} + 5000} = \frac{1}{0.001 (3 x^{2} + 5000)}$

Step 5.3.2.2

Cancel the common factor of $3 x^{2} + 5000$ .

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Step 5.3.2.2.1

Cancel the common factor.

$\frac{x' (3 x^{2} + 5000)}{3 x^{2} + 5000} = \frac{1}{0.001 (3 x^{2} + 5000)}$

Step 5.3.2.2.2

Divide $x'$ by $1$ .

$x' = \frac{1}{0.001 (3 x^{2} + 5000)}$

Step 5.3.3

Simplify the right side.

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Step 5.3.3.1

Rewrite $1$ as $1 (1)$ .

$x' = \frac{1 (1)}{0.001 (3 x^{2} + 5000)}$

Step 5.3.3.2

Separate fractions.

$x' = \frac{1}{0.001} \cdot \frac{1}{3 x^{2} + 5000}$

Step 5.3.3.3

Divide $1$ by $0.001$ .

$x' = 1000 \frac{1}{3 x^{2} + 5000}$

Step 5.3.3.4

Combine $1000$ and $\frac{1}{3 x^{2} + 5000}$ .

$x' = \frac{1000}{3 x^{2} + 5000}$

Step 6

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

$\frac{d x}{d C} = \frac{1000}{3 x^{2} + 5000}$