Find the derivatives of the functions in Exercises 1–42.

Question

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𝔂 = ( √ x )²

( 1 + x )

Accepted Answer

Welcome back everyone to another video. Calculate the derivative of the function G of Y equals square root of Y divided by Y + 5 raise to the power of 5. So essentially we want to identify the derivative, right, and we have to understand that G of Y is a composite function. It consists of our rational function square root of Y divided by Y + 5, and it is raised to the power of 5. So what we're going to do is simply define U of Y as the inner function square root of Y divided by Y + 5. And now we can say that G of U is simply equal to you to the power of 5, right? So if we want to differentiate G of Y, we essentially have to find the derivative of G with respect to you and then multiply. By the derivative of you with respect toy. This is what the chain rule says, right? So we're going to say that this is G with respect to you multiplied by the u derivative with respect toy. This is our chain rule. And now first rule we're going to say that the derivative of u to the power of 5 is 5u to the power of 4, right? This is based on the power rule. And now we want to identify the derivative of the. Rational function So let's apply the quotient rule because we have a numerator and a denominator, and the quotient rule says that first of all, we are taking the derivative of the numerator, we can rewrite it as Y, it's the power of 12. We're multiplying by. The denominator, which is Y + 5, we are subtracting our numeratory to the power of 1/2 multiplied by the derivative of the denominator Y + 5, and we are dividing everything by the square of the denominator Y + 52. So now the derivative of y to the power of 1/2 is 1/2 Y to the power of -12, and we're multiplying that by Y + 5, and now we want to identify the derivative of Y + 5, which is equal to 1, right, because the derivative of Y is 1 and the derivative of 5 is 0. So we're simply subtracting Y to the power of 1/2. And now we are dividing by Y. Plus 5 squared. Now what we can do from here is simply factor out Y to the power of -12, so our U derivative is equal to. Why to the power of. -12. We can also factor out 1/2. And this leaves us with Y + 5 from the first part and now for the second part because we are subtracting Y to the power of 1/2, well, this means that we simply need to subtract to Y in recs because now when we multiply those, right, 12 to the power of -1/2 multiplied by -2Y essentially gives us -1. And then for Y, we get -12 plus 1. This gives us positive 1/2, so that's correct. And now if we write the denominator, we have Y + 5 squad. So now let's combine the like terms. We can also write 2 in the denominator. So let's begin with the denominator 2 Y + 5 squared, and now in the numerator. We have Y to the power of -12 multiplied by 5 minus Y. Well then, we have identified the derivative. So now what we want to do is simply go back into the chain rule. We can now replace you with the actual definition of the function. So U is square root of Y, which we can write as y to the power of 1/2. Divided by Y + 5, this is race to the power of 4, right? And now we are multiplying this result by the derivative of U with respect to Y, which is Y to the power of -15 multiplied by 5 minus Y. Divided by 2 Y + 5 squared. So now, if we erase our first part. To the power of 4, while applying the properties of exponents, we're going to get Y raise the power of 1/2 multiplied by 4. This is 2, and then our denominator becomes Y + 5 raise to the power of 4. And I'll notice that we are multiplying, right? So essentially, let's multiply. We're going to have 5 Y. Squared multiplied by the power of -12. This gives us ys the power of 2 minus 12, which is 3 halves. We then have 5 minus y as our factor, and then considering the denominator, we end up with 2 Y + 5 raise to the power of 4 + 2, which is 6, and this is how we end up with the final answer to this problem. Thank you for watching.

Find the derivatives of the functions in Exercises 1–42.
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𝔂 = ( √ x )²
( 1 + x )

Key Concepts

Derivative

Chain Rule

Power Rule

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