Derivatives Find the derivative of the following functions. Se...

Question

Derivatives Find the derivative of the following functions. See Example 2 of Section 3.2 for the derivative of √x.

s(t) = 4√t - 1/4t⁴+t+1

Accepted Answer

Hello. In this video, we are going to be calculating the derivative of the given function. The function given to us is G of Y equal to 5 multiplied by the square root of Y minus 1/5 multiplied by Y to the 5th power, plus Y minus 3. Now, in order to take this derivative, for the first thing we need to do is we want to rewrite every term of Y in an exponential form. This will allow us to rewrite the square root of Y to be Y raised to the power of 1/2. Now what we want to do is we want to go ahead and take the derivative of each term of G of Y. Now notice that the 1st, 2nd, and 3 terms of Y are in an exponential form. The 3rd term Y has an exponential value of 1. In order to take a derivative of an exponential value of Y, we will go ahead and use the power rule. The power rule is defined as the following. If we want to take the derivative of an exponential value, X raises to the power of N, what we will do is we will bring down the value of the exponent N and multiply this by X, raise to the power of N minus 1. Let's go ahead and identify the value of N for each of the terms. In the first term, Y has an exponential value of 12, so our N is going to be 1/2. For the second term, the exponential value of Y is 5, so N will equal to 5. And in the third term, the exponential value of Y is 1, so N is equal to 1. What we will go ahead and do is we will take the derivatives using the power rule and the value of N we just identified. So, G Y, which is the derivative of Y, will be 5 multiplied by the derivative of Y raise the power of 1/2. By the power rule, we bring down the value of N, which is 1/2, multiply this by Y, raise the power of 1/2 minus 1. Then we repeat this process with the next term. We will take 1/5 and multiply this by the derivative of Y to the power of 5. We bring down the value of N, which is 5, and multiply this by Y, raise to the power of 5 minus 1. We repeat this process with the last term. We will go ahead and add the derivative of Y race to the power of 1. That is going to be 1, multiplied by Y race to the power of 1 minus 1. Now, what about the derivative of the constant value -3? While by the constant rule of derivatives, if we take the derivative of a constant value C, this is going to be zero. The derivative of any constant is zero, meaning that we are going to subtract 0 at the end of the derivative. Now what we need to do is we need to simplify the derivative G Y. Now, for the first term, 5, multiplied by 1/2, will give us 5 divided by 2. And in the exponent, 1/2 minus 1 will give us -12. So we have 5 halves multiplied by Y raised to the power of -12. For the second term, 1/5 multiplied by 5 will simplified to be -1, and this will be multiplied by Y raises the power of 5 minus 1, which is 4. In the final term, We have 1 multiplied by Y, raised to the power of 1 minus 1. Now, 1 minus 1 gives us the value of 0. So we have 1 multiplied by the value, why erase the power of 0. But anything with an exponential value of 0 is just going to be 1. So the third term will simplify to be 1. Now, this is going to be the derivative of the original G of Y function, but we don't want to leave the derivative with any negative exponents. The first thing we can do to get rid of this negative exponent is move the value of Ya the power of -12 to the denominator of 5 halves, and if we rewrite that into the form with a square root, G of Y will equal to 5 divided by 2 multiplied by the square root of Y minus Y to the power of 4 + 1. This is going to be the derivative of the G of Y function given to us. And with that being said, the solution to this problem is going to be a. So, I hope this video helps you in understanding on how to use the derivative rules, and I will go ahead and see you all in the next video.

Derivatives Find the derivative of the following functions. See Example 2 of Section 3.2 for the derivative of √x.
s(t) = 4√t - 1/4t⁴+t+1

Key Concepts

Derivatives

Power Rule

Chain Rule

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