Chapter 3 Test, Question # 5

5) Find the derivative of the following function: y = 9^7x


Ok so first we should realize that there is a variable in the exponent. So, we need to bring the x variable down by using logarithmic properties.

Step #1: Take the ln (natural log) of both sides of the equation. The property states: lnb(x^n) = n lnbx. Then we have the equation: ln y = (7x)(ln9).

Step #2: Take the derivative of new equation. The derivate is: (1/y)(y') = (7)(ln9)

Step #3: Isolate or solve for the y' by multiplying both sides by y. Then we get: y' = 7(ln 9)(y')

Step #4: We're almost there! Now dont forget to replace the y with the original function. The final answer is y' = (7)(9^7x)(ln9). The answer was B!!!

Be careful in this problem! DONT USE THE POWER RULE! Choice number A was the answer if we had taken the power rule...which is wrong wrong wrong!

Hope that helped everyone! Good luck.