give examples of functions for these various derivatives

Question

So I just had two questions. The first one states "an exponential function for which the derivative is always negative.

I have no idea how to do this because I thought you couldn't have a negative exponent and therefore not have a negative derivative. I only know how to find functions using the power rules because the other rules confuse me. If you could explain what rule to use and why I would be super grateful.

 

The second one is: A function f such that f'''(x)=f(x)

 

so this one I have no idea because my teacher hasn't even taught us about third derivatives yet. I'm not sure how to find this so please help me...

Thank you so much for your help.

Answer 1

Hi Olivia.

 

1) "An exponential function for which the derivative is always negative" is calculus-speak for "An exponential function which is decreasing everywhere."

 

If you make any exponent negative, it will cause the function to decrease.

 

f(x) = e^-x

 

This function is always decreasing, because the bigger x gets, the smaller f(x) gets.

 

2) The function your teacher is referencing is

 

f(x) = e^x

 

The derivative of e^x is just e^x. Meaning the second derivative of e^x is also just e^x. Meaning the third derivative of e^x is also e^x. And so on and so on until the infinite derivative of e^x (which will also just be e^x).