Derivatives of products and quotients Find the derivative of t...

Question

Derivatives of products and quotients Find the derivative of the following functions by first expanding or simplifying the expression. Simplify your answers.

y = 12s³-8s²+12s/4s

Accepted Answer

Hello. In this video, we are going to be computing the derivative of the given function. The function that is given to us in this problem is H of U is equal to 18 6 minus 12 of 5, plus 6 U 4, all divided by 6 U. So, before we jump into taking the derivative, let's first see if we can simplify the given function. Currently this is a rational function. We have a denominator of 6u and a trinomial in the numerator, but notice something about the numerator. Every coefficient in the numerator has some multiple of 6, and every term in the numerator has some multiple of u. What we can do with the numerator is we can factor out a 6 U, and once we do that, we can eliminate the 6U in the denominator, thus simplifying the rational function. So, let's go ahead and start by factoring a 6u from the numerator. Factoring out a 6u will give us 6u multiplied by the quantity, 3 multiplied by u to the 5th, minus 2 multiplied by you to the 4th, plus U to the power of 3. And this will be divided by 6 U. Now that we have a multiple of 6u in both the numerator and denominator, we can reduce these to be 1, and that will simplify each of you to be 3, u to the 5th, minus 24th, plus u cubed. Now that we've simplified our HOU function, let's go ahead and take the derivative. Now, notice that every variable U in our function is in an exponential form. What this means is that we can take the derivative by using the power rule for derivatives. The power rule states that if we have some function in an exponential form, then its derivative will equal to the value of the exponent multiplied by X the power of N minus 1. So, if we have F of X equal to X to the power of N, its derivative will be N multiplied by X to the power of N minus 1. We move the value of the exponent to the front of the variable and reduce the current exponential value by 1. We will do that for you to the 5th, you to the 4th, and you to the power of 3. So, H of U will equal to 3 multiplied by the derivative of u to the 5th, by the power rule, that derivative will be 5 u to the 5 minus 1, which will be 4. Minus 2 multiplied by the derivative of the 4th, which will be 4, the power of 3, plus the derivative of you to the 3rd, which will be 3 U squared. Now what we need to do is we need to simplify the derivative. We will simplify by multiplying the coefficients together. 3 multiplied by 5 will give us 15, so our first term is 15 to the 4th. -2 x 4 will be -8, so we have -8 youth to the 3 power. Then plus 3 U squared. And this is going to be the derivative of the original HOU function. And with that being said, the solution to this problem is going to be D. So I hope this video helps you in understanding how to find the derivative of a function, and I'll go ahead and see you all in the next video.

Derivatives of products and quotients Find the derivative of the following functions by first expanding or simplifying the expression. Simplify your answers.
y = 12s³-8s²+12s/4s

Key Concepts

Derivatives

Product and Quotient Rules

Simplification of Expressions

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