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Numerade Educator

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Problem 27 Medium Difficulty

Find the derivative of the function.
$ r(t) = 10^{2 \sqrt {t}} $

Answer

To solve this problem we can use the formula
d/dx[ ln a^u(x)]= ln a * a^u(x)*u'(x).
So in our problem a = 10 and u(t)=2sqrt [t] . The answer will then be
r'(t) = ln (10 )* 10^(2sqrt[t])*t^-(1/2).

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Video Transcript

all right. So here we have an exponential function, but we have a composite function. We have a function inside of function, so we're going to use the chain rule. And remember that when you're finding the derivative of a basic exponential function like 10 to the X, it's going to be 10 to the X Times natural log 10. So if you're using the chain rule with that and you're finding the derivative of 10 to the F of X, it's going to be 10 to the F of X times natural log 10 times the derivative of F of X because that would be the derivative of the inside sweep driven of of the outside tens, derivative of the inside. Okay, now let's apply that rule to this problem. And first I'm going to change how the problem looks. I'm gonna write it as 10 times to tea or 10 to the power to teach of the 1/2 power. I like to write my square root tea as a 1/2 power to make it easier to differentiate. Okay, so for the derivative, we're gonna start with 10 to the power that's their 10 to the power to t to the 1/2 multiplied by natural log 10. So that takes care of this part of it. Now we need to multiply by the derivative of the inside. So it's going to be the derivative of two t two. The 1/2 bring down the 1/2 and you get one t to the and then subtract one from the power. So teach of the negative 1/2. Okay, At this point, we have all the parts and we're just going to simplify. So let's change our teacher the 1/2 power back to a square root and then r t to the negative 1/2 is going to be moved to the denominator and made a square root. So we have 10 ta to the power to square rooty times natural log 10 over the square root of tea.