Calculus Examples

$f (x) = \frac{e^{x}}{4}$ ; $g (x) = x^{2}$

Step 1

Set up the composite result function.

$g (f (x))$

Step 2

Evaluate $g (\frac{e^{x}}{4})$ by substituting in the value of $f$ into $g$ .

$g (\frac{e^{x}}{4}) = {(\frac{e^{x}}{4})}^{2}$

Step 3

Apply the product rule to $\frac{e^{x}}{4}$ .

$g (\frac{e^{x}}{4}) = \frac{{(e^{x})}^{2}}{4^{2}}$

Step 4

Multiply the exponents in ${(e^{x})}^{2}$ .

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Step 4.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$g (\frac{e^{x}}{4}) = \frac{e^{x \cdot 2}}{4^{2}}$

Step 4.2

Move $2$ to the left of $x$ .

$g (\frac{e^{x}}{4}) = \frac{e^{2 x}}{4^{2}}$

Step 5

Raise $4$ to the power of $2$ .

$g (\frac{e^{x}}{4}) = \frac{e^{2 x}}{16}$