Answer: 65
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Explanation:
We'll need to compute the difference quotient. In this case, we need to find what [tex]\frac{g(h+t)-g(t)}{h}[/tex] is equal to. It's called a difference quotient because there's a subtraction in the numerator (aka "difference") and we're dividing to form the quotient.
The idea is that as h approaches 0, then that expression I wrote will approach the derivative we're after. Keep in mind that h will technically never get to 0 itself. It only gets closer and closer.
Anyways, let's compute [tex]g(h+t)[/tex] first
[tex]g(t) = 5t^2+5t\\\\g(h+t) = 5(h+t)^2+5(h+t)\\\\g(h+t) = 5(h^2+2ht+t^2)+5(h+t)\\\\g(h+t) = 5h^2+10ht+5t^2+5h+5t\\\\[/tex]
Then we'll subtract off g(t)
[tex]g(h+t)-g(t) = (5h^2+10ht+5t^2+5h+5t) - (5t^2+5t)\\\\g(h+t)-g(t) = 5h^2+10ht+5t^2+5h+5t - 5t^2-5t\\\\g(h+t)-g(t) = 5h^2+10ht+5h\\\\[/tex]
A very important thing to notice: the terms that don't have any 'h's in them have been canceled out (eg: 5t^2 combined with -5t^2 added to 0). Why is this important? It's because we need to factor 'h' out and we'll have a pair of 'h's cancel like so
[tex]\frac{g(h+t)-g(t)}{h} = \frac{5h^2+10ht+5h}{h}\\\\\frac{g(h+t)-g(t)}{h} = \frac{h(5h+10t+5)}{h}\\\\\frac{g(h+t)-g(t)}{h} = 5h+10t+5\\\\[/tex]
The left hand side cannot have h = 0, or else we have a division by zero error. But if we approached 0 (not actually getting there), then the expression 5h+10t+5 will approach 5(0)+10t+5 = 10t+5
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In short: The derivative of [tex]5t^2+5t[/tex] is [tex]10t+5[/tex]
In terms of symbols, [tex]g ' (t) = 10t+5[/tex]
Later on in calculus, you'll learn a shortcut so you won't have to compute the difference quotient every time you need a derivative. Refer to the power rule for more information.
After we find the derivative, it's as straight forward as plugging in t = 6 to compute g ' (6)
[tex]g ' (t) = 10t+5\\\\g ' (6) = 10(6)+5\\\\g ' (6) = 60+5\\\\g ' (6) = 65\\\\[/tex]
Side note: This tells us that the slope of the tangent line is m = 65 when t = 6. In other words, this line is tangent to g(t) when t = 6, and this particular tangent line has slope m = 65.
