Question

simplify the function before differentiating.

f(x)=\frac{e^{x}+6 e^{9 x}}{e^{x}}

f(x)=quad \text { (simplify your answer.) }

Explanation:

Step1: Split the fraction

Using the rule \(\frac{a + b}{c}=\frac{a}{c}+\frac{b}{c}\), we have \(f(x)=\frac{e^{x}}{e^{x}}+\frac{6e^{9x}}{e^{x}}\).

Step2: Simplify each term

For the first term, \(\frac{e^{x}}{e^{x}} = 1\) (since \(a^{m}\div a^{n}=a^{m - n}\), here \(m=n = x\) and \(e^{x-x}=e^{0}=1\)). For the second term, using \(a^{m}\div a^{n}=a^{m - n}\), \(\frac{e^{9x}}{e^{x}}=e^{9x-x}=e^{8x}\). So \(f(x)=1 + 6e^{8x}\).

Answer:

\(1+6e^{8x}\)