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Numerade Educator

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Problem 11 Easy Difficulty

Differentiate the function.

$ F(t) = (\ln t)^2 \sin t $

Answer

$$F^{\prime}(t)=(\ln t)^{2} \cos t+\sin t \cdot 2 \ln t \cdot \frac{1}{t}=\ln t\left(\ln t \cos t+\frac{2 \sin t}{t}\right)$$

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Video Transcript

for this problem. We're wanting to graph the function. Um, in this case, the function is FFT is equal to the natural log of peace squared times The sign of team of the natural log of chief on the entire thing is squared so well, actually put the parentheses on the outside and that's time to sign of team. This is the problem that we're dealing with and we want to. Ultimately, our ultimate goal of this problem is to learn how to differentiate with log rhythmic functions. So what we'll do in this case is used the product rule. Um, so if the product rule, we take the first value and take its derivative. So when we do that, we'll end up getting the derivative of the natural log of teeth squared. So it'll be to natural log of tea times the inside because it'll be changeable. So times one over tea and then that's all going to be times the sign of team than what we have for the second portion is this is going to be plus, uh, the natural log of t squared. I'm the co sign of key since the derivative of scientist coastline key we see In both cases we can factor out the natural log of tea. So our final answer for this is going to be the natural log of tea times to sign T over tea, plus the natural log of tea coats. I'm t so ultimately, this is going to be our final answer. We factored out a natural log of tea here and here That ends up giving us two times the sine of t over tea and one natural log of T leftover times coastline T. This will be our derivative based on, um, using the product role and the chain role.