Calculus Examples

$e^{100 x} - 3^{x}$

Step 1

By the Sum Rule, the derivative of $e^{100 x} - 3^{x}$ with respect to $x$ is $\frac{d}{d x} [e^{100 x}] + \frac{d}{d x} [- 3^{x}]$ .

$\frac{d}{d x} [e^{100 x}] + \frac{d}{d x} [- 3^{x}]$

Step 2

Evaluate $\frac{d}{d x} [e^{100 x}]$ .

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Step 2.1

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 $g (x) = 100 x$ .

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Step 2.1.1

To apply the Chain Rule, set $u$ as $100 x$ .

$\frac{d}{d u} [e^{u}] \frac{d}{d x} [100 x] + \frac{d}{d x} [- 3^{x}]$

Step 2.1.2

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

$e^{u} \frac{d}{d x} [100 x] + \frac{d}{d x} [- 3^{x}]$

Step 2.1.3

Replace all occurrences of $u$ with $100 x$ .

$e^{100 x} \frac{d}{d x} [100 x] + \frac{d}{d x} [- 3^{x}]$

Step 2.2

Since $100$ is constant with respect to $x$ , the derivative of $100 x$ with respect to $x$ is $100 \frac{d}{d x} [x]$ .

$e^{100 x} (100 \frac{d}{d x} [x]) + \frac{d}{d x} [- 3^{x}]$

Step 2.3

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

$e^{100 x} (100 \cdot 1) + \frac{d}{d x} [- 3^{x}]$

Step 2.4

Multiply $100$ by $1$ .

$e^{100 x} \cdot 100 + \frac{d}{d x} [- 3^{x}]$

Step 2.5

Move $100$ to the left of $e^{100 x}$ .

$100 e^{100 x} + \frac{d}{d x} [- 3^{x}]$

Step 3

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

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Step 3.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- 3^{x}$ with respect to $x$ is $- \frac{d}{d x} [3^{x}]$ .

$100 e^{100 x} - \frac{d}{d x} [3^{x}]$

Step 3.2

Differentiate using the Exponential Rule which states that $\frac{d}{d x} [a^{x}]$ is $a^{x} \ln (a)$ where $a$ = $3$ .

$100 e^{100 x} - 3^{x} \ln (3)$

Step 4

Reorder terms.

$- 3^{x} \ln (3) + 100 e^{100 x}$