## Thursday, November 09, 2006

### Chapter 3 Test: Question #1

Problem #1: Differentiate y=8^x(sin(x)-cos(x))

Solution:
1) Before we find the derivative of the function, understand the rules that must be applied to find the derivative of this function: Product Rule

2) First, find the two functions that are being multiplied: 8 ^x and (sin(x)-cos(x))

3) Next, find the derivative of both the functions:
d/dx 8^x= d/dx e^((ln 8)x)= e^((ln 8)x) x d/dx(ln a)x= e^((ln 8)x)) x ln 8= 8^x ln 8
AND
(sin(x)-cos(x))=(cos(x)-(-sin x))=(cos(x)+sin(x))= ((sin(x)+cos(x))

4)Simply plug in the numbers that are required for the product rule:
8^x ln 8(sin(x)-cos(x))+8^x(sin(x)+cos(x)) = 8^x(sin(x)+cos(x)) + 8^x(sin(x)-cos(x))ln 8= Choice C

At 8:11 AM,  Math Maverick said...

Hi Joseph.
In finding the derivative of 8^x, is it really necessary to go through all that effort? One of our calculus concept sheets is the derivative of an exponential function (not e^x)...

Your derivative is correct, but this is a case where knowing the formula can save some significant time on the test.