Calculus Examples

$C x (1 - e^{- k x})$

Step 1

Since $C$ is constant with respect to $x$ , the derivative of $C x (1 - e^{- k x})$ with respect to $x$ is $C \frac{d}{d x} [x (1 - e^{- k x})]$ .

$C \frac{d}{d x} [x (1 - e^{- k x})]$

Step 2

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x$ and $g (x) = 1 - e^{- k x}$ .

$C (x \frac{d}{d x} [1 - e^{- k x}] + (1 - e^{- k x}) \frac{d}{d x} [x])$

Step 3

Differentiate.

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Step 3.1

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

$C (x (\frac{d}{d x} [1] + \frac{d}{d x} [- e^{- k x}]) + (1 - e^{- k x}) \frac{d}{d x} [x])$

Step 3.2

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

$C (x (0 + \frac{d}{d x} [- e^{- k x}]) + (1 - e^{- k x}) \frac{d}{d x} [x])$

Step 3.3

Add $0$ and $\frac{d}{d x} [- e^{- k x}]$ .

$C (x \frac{d}{d x} [- e^{- k x}] + (1 - e^{- k x}) \frac{d}{d x} [x])$

Step 3.4

Since $- 1$ is constant with respect to $x$ , the derivative of $- e^{- k x}$ with respect to $x$ is $- \frac{d}{d x} [e^{- k x}]$ .

$C (x (- \frac{d}{d x} [e^{- k x}]) + (1 - e^{- k x}) \frac{d}{d x} [x])$

Step 4

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

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Step 4.1

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

$C (x (- (\frac{d}{d u} [e^{u}] \frac{d}{d x} [- k x])) + (1 - e^{- k x}) \frac{d}{d x} [x])$

Step 4.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u} [a^{u}]$ is $a^{u} \ln (a)$ where $a$ = $e$ .

$C (x (- (e^{u} \frac{d}{d x} [- k x])) + (1 - e^{- k x}) \frac{d}{d x} [x])$

Step 4.3

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

$C (x (- (e^{- k x} \frac{d}{d x} [- k x])) + (1 - e^{- k x}) \frac{d}{d x} [x])$

Step 5

Differentiate.

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Step 5.1

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

$C (x (- e^{- k x} (- k \frac{d}{d x} [x])) + (1 - e^{- k x}) \frac{d}{d x} [x])$

Step 5.2

Multiply.

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Step 5.2.1

Multiply $- 1$ by $- 1$ .

$C (x (1 e^{- k x} (k \frac{d}{d x} [x])) + (1 - e^{- k x}) \frac{d}{d x} [x])$

Step 5.2.2

Multiply $e^{- k x}$ by $1$ .

$C (x (e^{- k x} (k \frac{d}{d x} [x])) + (1 - e^{- k x}) \frac{d}{d x} [x])$

Step 5.3

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

$C (x (e^{- k x} (k \cdot 1)) + (1 - e^{- k x}) \frac{d}{d x} [x])$

Step 5.4

Multiply $k$ by $1$ .

$C (x (e^{- k x} k) + (1 - e^{- k x}) \frac{d}{d x} [x])$

Step 5.5

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

$C (x (e^{- k x} k) + (1 - e^{- k x}) \cdot 1)$

Step 5.6

Multiply $1 - e^{- k x}$ by $1$ .

$C (x (e^{- k x} k) + 1 - e^{- k x})$

Step 6

Simplify.

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Step 6.1

Apply the distributive property.

$C (x (e^{- k x} k)) + C \cdot 1 + C (- e^{- k x})$

Step 6.2

Multiply $C$ by $1$ .

$C x (e^{- k x} k) + C + C (- e^{- k x})$

Step 6.3

Reorder terms.

$C + e^{- k x} C k x - e^{- k x} C$

Step 6.4

Reorder factors in $C + e^{- k x} C k x - e^{- k x} C$ .

$C + C k x e^{- k x} - C e^{- k x}$