I am confused on how to simplify my derivative

Question

Find the derivative of the given function. (Remember to simplify the numerator of your answer.)
z(t) = (t^2+5t+2)/ t+7

I found the derivative and then I simplified my answer to

((t+3)(t+11))/(t+7)^2  but I got it wrong. I am pretty sure I did it right but I don't know how to find where I went wrong. please correct me, and help me through the process. Thank you.

Answer 1

z(t) = (t² + 5t + 2) / (t + 7)  [NOTE:  Please include the parenthesis so there is no doubt what the denominator is.]

 

f(t) = t² + 5t + 2

g(t) = t + 7

 

z(t) = f(t) / g(t)

 

Apply the quotient rule:

 

z'(t) = [f'(t)g(t) - f(t)g'(t)] / [g(t)]²

 

z'(t) = [(2t + 5)(t + 7) - (t² + 5t + 2)(1)] / (t + 7)²

 

The numerator gives:

 

2t² + 14t + 5t + 35 - (t² + 5t + 2) = t² + 14t + 33 = (t + 11)(t + 3)

 

Then z'(t) = (t + 11)(t + 3) / (t + 7)²

Answer 2

This is another instance of the Quotient Rule.

 

If f = t² + 5t + 2

and g = t + 7

 

Our derivative is [g * f' - (f * g')] ÷ (g)²

 

f' = 2t + 5

g' = 1

(g)² = (t + 7)² 

 

So this derivative is [(t + 7)(2t + 5) - (t² + 5t + 2)(1)] ÷ (t + 7)²

 

The numerator is:

 

2t² + 5t + 14t + 35 - t² - 5t - 2 = t² + 14t  + 33 = (t + 3)(t + 11)

 

The denominator is still (t + 7)²

 

Note that for this quotient, t cannot equal -7 because that would give us 0 in the denominator, and division by 0 is undefined.