9–61. Evaluate and simplify y'.

y=sin √c...

Question

9–61. Evaluate and simplify y'.

y=sin √cos² x+1

Accepted Answer

Welcome back, everyone. In this problem, we want to find the derivative of the function Y equals the cosine of the square root of sine squared X + 4. Now, how are we going to differentiate this function? Notice that Y is a composite function, OK? It has an inner, an inner function, the square root of sin square x + 4, and an alter function, the cosine of the square root of sin squared X + 4, right? Now, if we have a composite function, we can differentiate composite functions using the chain rule of differentiation. Recall that the chain rule basically says that if we have a function y that's equal to U to the power of n where U is a function of X. Then the derivative of y with respect to X will be equal to the derivative of Y with respect to u multiplied by the derivative of u with respect to X. Nor how can we apply that to our problem. Well, for our problem, we can let you be equal to our inner function. The square root of sin square x + 4, OK. And now in that case. Then that means Y is going to be equal to the cosine of you. So now if we can differentiate you with respect to X and Y with respect to you, then we should be able to plug it into our. Chain rule to help us solve for the derivative of Y with respect to X. Now, let's start with you, OK? Now we could also rewrite you as the sign of X2 + 4 raised to a half, and that might make it a bit easier to read. No, we can differentiate you with respect to X. Notice that U itself is another composite function, so we can apply the same idea. But I'm just going to write it in a different way so that we don't get too confused as we go along. Now another way we can write the chain rule, OK, is to say that for you, OK, being equal to. Or rather, let's again, let's just talk in general terms here. For Y be being equal to U raises to the power of N, then the derivative of Y with respect to X is gonna be equal to N. We bring down the power. Next we multiply it by u to the n minus 1. We subtract one from the poor in our term and then we multiply it by the derivative of our inner function U. So if we apply that idea here to our function, OK, then the derivative of you with respect to X. We can find it by first bringing down the power. That's gonna be a 1/2 multiplied by our original term sine squared X + 4 raised to the power of 1/2 minus 1, which is negative 1/2 multiplied by the derivative of our inner function sine squared X + 4. And the derivative of sin squared X, OK, is gonna be equal to, it's gonna be equal to 2 sin x plus X, OK. Where the derivative of 4 is equal to 0, so we're just left with 2 sin x + X multiplied by this entire expression. Now can we make this any simpler? Well, we also know that we can use the double angle formula in trigonometry here. And that formula basically tells us that the sin of 2 X is equal, OK, to 2 sin X. Multiplied by the cosine of X, OK. So, if we use that idea, if we apply that idea here, then we could take it a step further and rewrite this expression. OK. As a half of sine squared X + 4 raised to the point of negative 5 multiplied by 22 X. OK? Sign to X. I know we could rewrite uh sine sine square x + 4 is to the power of negative in our denominator. So now when we simplify this, it is going to be the sign of 2 X, OK. Divided by. To multiply by the square root of sine squared X + 4. Thus, this is the derivative of our inner function with respect to X. OK, so let's keep that in mind. Next, there's differentiatey with respect to you. Now if y equals the cosine of you, that means the derivative of y with respect to you will be equal to negative sine U, which we know our expression u. So that's really negative sign multiplied by the square root of sine squared X. Plus 4 Now that we have the derivatives of the inner and outer functions, we can apply the chain rule, OK? So using what we know here, OK. This implies then. Let me put it in black actually. This would imply that DUIDX. Eals DUIDU. So again, we know what that is negative sign. The negative sign of square root of sin squared X + 4, OK. Multiplied by DUDX, the sin of 2 X, OK, divided by 2 multiplied by the square root of sine squared X plus 4. And know if we go ahead and simplify that a bit further, clean up our expression, then we're going to get the negative value of sin to X. Multiplied by the sign of the square root of sin squared X + 4. OK. Now all of this is being divided by 2 multiplied by the square root of sine squared X + 4. Thus, this is the derivative of Y with respect to X. Thanks a lot for watching everyone. I hope this video helped.

9–61. Evaluate and simplify y'.

y=sin √cos² x+1

Key Concepts

Chain Rule

Trigonometric Derivatives

Implicit Differentiation

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