Calculus Examples

$f (x) = 6 x^{2} e (\frac{6}{25} x) - 50 x e (\frac{6}{25} x) + 208.33 e^{\frac{6}{25}}$

Step 1

By the Sum Rule, the derivative of $6 x^{2} e (\frac{6}{25} x) - 50 x e (\frac{6}{25} x) + 208.33 e^{\frac{6}{25}}$ with respect to $x$ is $\frac{d}{d x} [6 x^{2} e (\frac{6}{25} x)] + \frac{d}{d x} [- 50 x e (\frac{6}{25} x)] + \frac{d}{d x} [208.33 e^{\frac{6}{25}}]$ .

$\frac{d}{d x} [6 x^{2} e (\frac{6}{25} x)] + \frac{d}{d x} [- 50 x e (\frac{6}{25} x)] + \frac{d}{d x} [208.33 e^{\frac{6}{25}}]$

Step 2

Evaluate $\frac{d}{d x} [6 x^{2} e (\frac{6}{25} x)]$ .

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Step 2.1

Combine $\frac{6}{25}$ and $x$ .

$\frac{d}{d x} [6 x^{2} e \frac{6 x}{25}] + \frac{d}{d x} [- 50 x e (\frac{6}{25} x)] + \frac{d}{d x} [208.33 e^{\frac{6}{25}}]$

Step 2.2

Combine $6$ and $\frac{6 x}{25}$ .

$\frac{d}{d x} [x^{2} e \frac{6 (6 x)}{25}] + \frac{d}{d x} [- 50 x e (\frac{6}{25} x)] + \frac{d}{d x} [208.33 e^{\frac{6}{25}}]$

Step 2.3

Multiply $6$ by $6$ .

$\frac{d}{d x} [x^{2} e \frac{36 x}{25}] + \frac{d}{d x} [- 50 x e (\frac{6}{25} x)] + \frac{d}{d x} [208.33 e^{\frac{6}{25}}]$

Step 2.4

Combine $\frac{36 x}{25}$ and $x^{2}$ .

$\frac{d}{d x} [\frac{36 x \cdot x^{2}}{25} e] + \frac{d}{d x} [- 50 x e (\frac{6}{25} x)] + \frac{d}{d x} [208.33 e^{\frac{6}{25}}]$

Step 2.5

Multiply $x$ by $x^{2}$ by adding the exponents.

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Step 2.5.1

Move $x^{2}$ .

$\frac{d}{d x} [\frac{36 (x^{2} x)}{25} e] + \frac{d}{d x} [- 50 x e (\frac{6}{25} x)] + \frac{d}{d x} [208.33 e^{\frac{6}{25}}]$

Step 2.5.2

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

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Step 2.5.2.1

Raise $x$ to the power of $1$ .

$\frac{d}{d x} [\frac{36 (x^{2} x^{1})}{25} e] + \frac{d}{d x} [- 50 x e (\frac{6}{25} x)] + \frac{d}{d x} [208.33 e^{\frac{6}{25}}]$

Step 2.5.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{d}{d x} [\frac{36 x^{2 + 1}}{25} e] + \frac{d}{d x} [- 50 x e (\frac{6}{25} x)] + \frac{d}{d x} [208.33 e^{\frac{6}{25}}]$

Step 2.5.3

Add $2$ and $1$ .

$\frac{d}{d x} [\frac{36 x^{3}}{25} e] + \frac{d}{d x} [- 50 x e (\frac{6}{25} x)] + \frac{d}{d x} [208.33 e^{\frac{6}{25}}]$

Step 2.6

Combine $\frac{36 x^{3}}{25}$ and $e$ .

$\frac{d}{d x} [\frac{36 x^{3} e}{25}] + \frac{d}{d x} [- 50 x e (\frac{6}{25} x)] + \frac{d}{d x} [208.33 e^{\frac{6}{25}}]$

Step 2.7

Since $\frac{36 e}{25}$ is constant with respect to $x$ , the derivative of $\frac{36 x^{3} e}{25}$ with respect to $x$ is $\frac{36 e}{25} \frac{d}{d x} [x^{3}]$ .

$\frac{36 e}{25} \frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 50 x e (\frac{6}{25} x)] + \frac{d}{d x} [208.33 e^{\frac{6}{25}}]$

Step 2.8

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

$\frac{36 e}{25} (3 x^{2}) + \frac{d}{d x} [- 50 x e (\frac{6}{25} x)] + \frac{d}{d x} [208.33 e^{\frac{6}{25}}]$

Step 2.9

Combine $3$ and $\frac{36 e}{25}$ .

$\frac{3 (36 e)}{25} x^{2} + \frac{d}{d x} [- 50 x e (\frac{6}{25} x)] + \frac{d}{d x} [208.33 e^{\frac{6}{25}}]$

Step 2.10

Multiply $36$ by $3$ .

$\frac{108 e}{25} x^{2} + \frac{d}{d x} [- 50 x e (\frac{6}{25} x)] + \frac{d}{d x} [208.33 e^{\frac{6}{25}}]$

Step 2.11

Combine $\frac{108 e}{25}$ and $x^{2}$ .

$\frac{108 e x^{2}}{25} + \frac{d}{d x} [- 50 x e (\frac{6}{25} x)] + \frac{d}{d x} [208.33 e^{\frac{6}{25}}]$

Step 3

Evaluate $\frac{d}{d x} [- 50 x e (\frac{6}{25} x)]$ .

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Step 3.1

Combine $\frac{6}{25}$ and $x$ .

$\frac{108 e x^{2}}{25} + \frac{d}{d x} [- 50 x e \frac{6 x}{25}] + \frac{d}{d x} [208.33 e^{\frac{6}{25}}]$

Step 3.2

Combine $- 50$ and $\frac{6 x}{25}$ .

$\frac{108 e x^{2}}{25} + \frac{d}{d x} [x e \frac{- 50 (6 x)}{25}] + \frac{d}{d x} [208.33 e^{\frac{6}{25}}]$

Step 3.3

Multiply $6$ by $- 50$ .

$\frac{108 e x^{2}}{25} + \frac{d}{d x} [x e \frac{- 300 x}{25}] + \frac{d}{d x} [208.33 e^{\frac{6}{25}}]$

Step 3.4

Combine $\frac{- 300 x}{25}$ and $x$ .

$\frac{108 e x^{2}}{25} + \frac{d}{d x} [\frac{- 300 x \cdot x}{25} e] + \frac{d}{d x} [208.33 e^{\frac{6}{25}}]$

Step 3.5

Raise $x$ to the power of $1$ .

$\frac{108 e x^{2}}{25} + \frac{d}{d x} [\frac{- 300 (x^{1} x)}{25} e] + \frac{d}{d x} [208.33 e^{\frac{6}{25}}]$

Step 3.6

Raise $x$ to the power of $1$ .

$\frac{108 e x^{2}}{25} + \frac{d}{d x} [\frac{- 300 (x^{1} x^{1})}{25} e] + \frac{d}{d x} [208.33 e^{\frac{6}{25}}]$

Step 3.7

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{108 e x^{2}}{25} + \frac{d}{d x} [\frac{- 300 x^{1 + 1}}{25} e] + \frac{d}{d x} [208.33 e^{\frac{6}{25}}]$

Step 3.8

Add $1$ and $1$ .

$\frac{108 e x^{2}}{25} + \frac{d}{d x} [\frac{- 300 x^{2}}{25} e] + \frac{d}{d x} [208.33 e^{\frac{6}{25}}]$

Step 3.9

Combine $\frac{- 300 x^{2}}{25}$ and $e$ .

$\frac{108 e x^{2}}{25} + \frac{d}{d x} [\frac{- 300 x^{2} e}{25}] + \frac{d}{d x} [208.33 e^{\frac{6}{25}}]$

Step 3.10

Cancel the common factor of $- 300$ and $25$ .

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Step 3.10.1

Factor $25$ out of $- 300 x^{2} e$ .

$\frac{108 e x^{2}}{25} + \frac{d}{d x} [\frac{25 (- 12 x^{2} e)}{25}] + \frac{d}{d x} [208.33 e^{\frac{6}{25}}]$

Step 3.10.2

Cancel the common factors.

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Step 3.10.2.1

Factor $25$ out of $25$ .

$\frac{108 e x^{2}}{25} + \frac{d}{d x} [\frac{25 (- 12 x^{2} e)}{25 (1)}] + \frac{d}{d x} [208.33 e^{\frac{6}{25}}]$

Step 3.10.2.2

Cancel the common factor.

$\frac{108 e x^{2}}{25} + \frac{d}{d x} [\frac{25 (- 12 x^{2} e)}{25 \cdot 1}] + \frac{d}{d x} [208.33 e^{\frac{6}{25}}]$

Step 3.10.2.3

Rewrite the expression.

$\frac{108 e x^{2}}{25} + \frac{d}{d x} [\frac{- 12 x^{2} e}{1}] + \frac{d}{d x} [208.33 e^{\frac{6}{25}}]$

Step 3.10.2.4

Divide $- 12 x^{2} e$ by $1$ .

$\frac{108 e x^{2}}{25} + \frac{d}{d x} [- 12 x^{2} e] + \frac{d}{d x} [208.33 e^{\frac{6}{25}}]$

Step 3.11

Since $- 12 e$ is constant with respect to $x$ , the derivative of $- 12 x^{2} e$ with respect to $x$ is $- 12 e \frac{d}{d x} [x^{2}]$ .

$\frac{108 e x^{2}}{25} - 12 e \frac{d}{d x} [x^{2}] + \frac{d}{d x} [208.33 e^{\frac{6}{25}}]$

Step 3.12

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

$\frac{108 e x^{2}}{25} - 12 e (2 x) + \frac{d}{d x} [208.33 e^{\frac{6}{25}}]$

Step 3.13

Multiply $2$ by $- 12$ .

$\frac{108 e x^{2}}{25} - 24 e x + \frac{d}{d x} [208.33 e^{\frac{6}{25}}]$

Step 4

Evaluate $\frac{d}{d x} [208.33 e^{\frac{6}{25}}]$ .

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Step 4.1

Multiply $208.33$ by $e^{\frac{6}{25}}$ .

$\frac{108 e x^{2}}{25} - 24 e x + \frac{d}{d x} [264.83933548]$

Step 4.2

Since $264.83933548$ is constant with respect to $x$ , the derivative of $264.83933548$ with respect to $x$ is $0$ .

$\frac{108 e x^{2}}{25} - 24 e x + 0$

Step 5

Add $\frac{108 e x^{2}}{25} - 24 e x$ and $0$ .

$\frac{108 e x^{2}}{25} - 24 e x$