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I am looking for the instantaneous growth rate of a function and having trouble solving it by hand.

This is the question:

$$f(x)= \dfrac{x}{e^{3x}}$$

So I convert $\dfrac{x}{e^{3x}}$ into $xe^{-3x}$

I then use the average growth rate formula to solve for the Instantaneous growth rate $$\dfrac{f(x+h) - f(x)}{h}$$

I get (x+h)e^-3(x+h)/h - xe^-3x/h

I distribute the e^-3(x+h) to the (x+h) and end up with:

xe^-3(x+h)+he^-3(x+h) / h - xe^-3x/h

I then distribute the -3 to the variables in the exponents:

(xe^-3x * xe^-3h) + (he^-3x * he^-3h) / h - (xe^-3x)/h

This is where I am stuck. Where do I go from here? I cannot seem to get the right answer.

I have tried setting h to almost 0 and can setup

(xe^3x * xe^-3h) + (he^-3x * h)-(xe^-3x) / h

Where do I go from there?

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    Why not just apply the product rule to $x \cdot e^{-3x}$ -- or are you required to apply the $f(x+h)-f(x)$ etc. method? – 2017-02-07
  • 0
    We have not covered the product rule yet. So I am required to apply the f(x+h)−f(x) method. – 2017-02-07

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