27–76. Calculate the derivative of the following functions.

Question

y = √f(x), where f is differentiable and nonnegative at x.

Accepted Answer

Hello there. Today we're going to solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. For the given function determine the derivative, assuming that F is differentiable and X is greater than 0. Y is equal to 1 divided by the square root of F of X. Awesome. So it appears for this particular problem, we're asked to figure out what the derivative is for this specific function, or rather this equation. Because we're trying to figure out the derivative of this particular function for Y, assuming that F is differentiable and X is greater than 0. So with that in mind, first off, in order to solve for this problem, we need to recall a note that in order to calculate the derivative of this function of Y, we will need to use the chain rule and the power rule. So we need to recall a note. That the chain rule states that if we have a composite function, which, let's write this down, if we have a composite function of G of F of X, Then its derivative will be G of F of X. Multiplied by F of X. And let us also recall and note that the power rule states that the derivative of X ton power is N multiplied by X of N minus 1. Awesome. So now our next step is we need to rewrite the function in a form that makes it easier to differentiate. So we could take our original function, which is Y equals 1 divided by the square root of F of X, and we could write it as, instead, let's draw an arrow, we could write this this function as Y equals F of X to the power of negative 1/2. Awesome. So now our next step is we need to, once again, we need to recall and apply the power rule. So the derivative of Y equals F of X to the power of -12 with respect to F of X will be -12 multiplied by F of X to the power of -3 halves. Fantastic. So moving right along here, our next step now is we need to recall and apply the chain rule. So we need to multiply the derivative from our second step, which we just performed, which is with respect to F of X. So let's, so just in case we get confused, let's label it. So this was step 1, this is step 2. So our third step now. we need to apply the chain rule, once again, so we need to multiply the derivative from step 2 by the derivative of the inner function of F of X, which, as we should recall, F of X. Is F of X. So our fourth step now. we need to combine the steps to write our final derivative, so the derivative of Y, so now we're saying the derivative of Y, with respect to. Once again, 2 X is -12 multiplied by F of X to the power of -3 halves multiplied by F of X. So therefore, the derivative of the derivative of our function for Y will be D Y by DX is equal to negative. 1/2 multiplied by F of X to the power of negative 3 halves multiplied by F of X, and that's it. That is our final answer for the derivative. Hooray, we did it. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

27–76. Calculate the derivative of the following functions.
y = √f(x), where f is differentiable and nonnegative at x.

Key Concepts

Derivative

Chain Rule

Square Root Function

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