Composition containing sin x Suppose f is differentiable on...

Question

Composition containing sin x Suppose f is differentiable on [−2,2] with f′(0)=3 and f′(1)=5. Let g(x)=f(sin x). Evaluate the following expressions.

c. g'(π)

Accepted Answer

Welcome back, everyone. Let HX be equal to PAX, where P is a differentiable function and the interval open bracket -3, 3 closed bracket with P 0 equals -4 and P 1 equals 2, where just means that it's the derivative. Calculate H of pi divided by 2. A says it's -4, B4, C -2, and the D2. Now if we're going to calculate H of pi divided by 2, we'll first need to find the derivative of H. In other words, H X. So let's try to do that. Let's first try to differentiate H of X. With respect to X, OK. Now, notice here that H of X is a composite function, OK, where X is the inner function and PO of X is the other function. OK. So no. That means that H of X, OK, is going to be equal to the derivative of the auto function by the chain rule, which is P of the cosine of X multiplied by the derivative of the inner function, and the derivative of the cosine of X equals negative sin of X, OK? So this is the derivative of H with respect to X. So no evaluating at X equals pi divided by 2. Then when we substitute, that means each prime of pi divided by 2 is equal to. P of the cosine of pi divided by 2. Multiplied by a negative sign of pi divided by 2. Now what do we know about the cosine and sine of pi divided by 2. Well, we know that the cosine of pi divided by 2 equals 0, OK. And the sin of pi divided by 2 equals 1. So really, then H of pi divided by 2 is going to be equal to P 0, OK, multiplied by -1. Remember earlier we know that P 0 equals -4 from the problem statement. So this is going to be equal to -4 multiplied by -1, which equals 4. Thus, each prime of I divided by 2 is equal to 4. B is the correct answer. Thanks a lot for watching everyone. I hope this video helped.

​Composition containing si​n​ x Suppose f is differentiable on [−2,2] with f′(0)=3 and f′(1)=5. Let g(x)=f(sin x). Evaluate the following expressions.c. g'(π)

Watch next

Composition containing sin x Suppose f is differentiable on [−2,2] with f′(0)=3 and f′(1)=5. Let g(x)=f(sin x). Evaluate the following expressions.
c. g'(π)