SOLUTION: Using the function y=(4x^2-8x+3)^4 find y' using the chain rule(extended power rule)answer in the form y'=A(Bx^2-Cx+D)^e(fx-G) note all answers are integers which I got what is

Algebra ->  Exponential-and-logarithmic-functions -> SOLUTION: Using the function y=(4x^2-8x+3)^4 find y' using the chain rule(extended power rule)answer in the form y'=A(Bx^2-Cx+D)^e(fx-G) note all answers are integers which I got what is       Log On


   

Question 201612: Using the function
y=(4x^2-8x+3)^4 find y' using the chain rule(extended power rule)answer in the form y'=A(Bx^2-Cx+D)^e(fx-G)
note all answers are integers
which I got what is 4(4x^2-8x+3)^3 but can not figure out (fx-g) part and then it ask now use y' to find the equation of the tangent line at the point (2,81)
y=_________________ x-_____________
thanks
Answer by jsmallt9(3759) About Me  (Show Source):
You can put this solution on YOUR website!
(Because of some limitations of the software on the Algebra.com site, I am doing to use dy/dx instead of y'.)
Let u+=+4x%5E2+-8x+%2B+3 then %28du%29%2F%28dx%29+=+8x+-+8
Substituting u into your equation we get
y+=+u%5E4
And, using the chain rule, we get
%28dy%29%2F%28dx%29+=+4%2Au%5E3%2A%28%28du%29%2F%28dx%29%29
Substituting back in for both u and du/dx we get:
%28dy%29%2F%28dx%29+=+4%284x%5E2+-8x+%2B+3%29%5E3%2A%288x+-+8%29

To find the equation of a tangent line (or any line) we need, in general, a point on the line and the slope for the line. We have the point, (2, 81). We need the slope. But y' gives us a formula for finding the slope of the tangent. We can use it to find the slope at x=2:
%28dy%29%2F%28dx%29+=+4%284%282%29%5E2+-8%282%29+%2B+3%29%5E3%2A%288%282%29+-8%29
Squaring
%28dy%29%2F%28dx%29+=+4%284%284%29+-8%282%29+%2B+3%29%5E3%2A%288%282%29+-8%29
Multiplying inside both parentheses
%28dy%29%2F%28dx%29+=+4%2816+-16+%2B+3%29%5E3%2A%2816+-8%29
Adding and subtracting inside both parentheses
%28dy%29%2F%28dx%29+=+4%283%29%5E3%2A%288%29
Cubing 3
%28dy%29%2F%28dx%29+=+4%2827%29%2A%288%29
Multiplying
%28dy%29%2F%28dx%29+=+864

Now we can use either the point-slope form y+-+y%5B1%5D+=+m%2A%28x+-+x%5B1%5D%29 or the slope intercept form y+=+mx+%2B+b to write the equation of the line through (2, 81) with a slope of 864. Using the point slope form we get:
y+-+81+=+864%28x+-+2%29
If you need to have your answer in slope-intercept form, multiply
y+-+81+=+864x+-+1728%29
Then add 81
y+=+864x+-+1647
or
y+=+864x+%2B+%28-1647%29