derivatives

Question

f(x)= (5x² -7)(x²-2) / x²+6

find f ' (x)  at x=5

 

 

Answer 1

y(x^2+6) = (5x^2-7)(x^2-2)

Differentiate both sides with respect to x,

y' (x^2+6) + y(2x) = 10x(x^2-2) + 2x(5x^2-7)

Since f(5) = 2714/31, plug in x = 5 and y = 2714/31, and solve for y',

y' = 46.920 <==Answer

Answer 2

First use the quotient rule.

 

f ' (x) = [((x)² + 6)[(5(x)²-7)((x)² - 2)]' - [(5(x)²-7)((x)² - 2)]((x)² + 6)' ]/((x)² + 6)²

 

Now we need to know derivatives of g(x) = (5(x)² - 7)((x)² - 2) and h(x) = (x)² + 6.

 

Clearly h ' (x) = 2(x) and for the derivative of g(x) we use the product rule.  g ' (x) = (10x)((x)²-2) + (5(x)² - 7)(2x) 

 

So putting this together we get f ' (x) = [((x)² + 6)[(10(x))((x)² - 2) + (5(x)² - 7)(2x)] - [(5(x)²-7)((x)² - 2)](2x)]/((x)² + 6)²

 

Plugging in 5 for x in the last equation gives 46.9199

 

So to sum up, we applied a quotient rule first to the whole thing and within that we used the product rule for the derivative of the numerator.

 

Hope this helps.

 

Philip