Derivatives Find and simplify the derivative of the following ...

Question

Derivatives Find and simplify the derivative of the following functions.

g(t) = t³+3t²+t / t³

Accepted Answer

Hello. In this video, we are going to be determining the derivative of G of X. The G of X function given to us is G X equal to 4X4 plus 3X cubed minus 2 X2 all divided by X the power of 4. So, before we decide to take the derivative, let's see if we can first simplify the G of X function. Notice that every term in the numerator has a multiple of X squared. So the first thing we can do is we can factor out an X2d from G of X. That will allow us to rewrite G of X as X2 multiplied by the quantity for X2 + 3X minus 2 divided by X 4th power. Next, what we can do is we can reduce a factor of X2d from both the numerator and denominator. That will allow us to reduce X2 to be 1, and reduce the exponent of X the 4th to be X2. So the simplified form of our G of X function will be 4 X2 plus 3X minus 2, divided by X2. This is going to be the simplified form of G of X, but how do we take the derivative of G of X? Notice that G of X is in the form of a rational function. If we want to take the derivative of rational function, we will need to use the quotient rule. The quotient rule is defined as the following. If we want to take the derivative of rational function, we will define the numerator and denominator as the respective functions F of X and G of X. Then the derivative will be defined. As G of X multiplied by the derivative F of X, plus F of X multiplied by the derivative G of X, divided. By the original G of X squared. So, in order to use the quotient rule, let's go ahead and define our F of X and G of X function for the quotient rule. F of X will be the function in the numerator, so F of X will equal to 4 X 2 plus 3X minus 2. G of X will be the function in the denominator, and that will be X squared. Next, we need the derivatives of both the numerator and denominator. The derivative of the numerator can be found by using the power rule for the numerator, and that will give us F of X equal to 8 X plus 3. For G of X, in order to take the derivative, we will also use the power rule. G of X will equal to 2 X. Now that we have the pieces of information we need, we can now start to use the quotient rule. So, the quotient rule will be defined as G of X, which will be X2, multiplied by F of X, which is 8 X + 3 minus F of X, which is 4 X2 + 3 X minus 2 multiplied by the derivative of G of X, which will be 2 X, divided by G of X2. Which will be X2 quantity squared. Now what we need to do is we need to simplify our derivative. We will start by distributing X2 to 8 X + 3. We will also distribute 2 X to 4X2 plus 3X minus 2. And X2 rates the power of 2 will simplify to be X to the power of 4. So, the derivative will simplify to be 8 X cubed plus 3X quad minus the quantity 8X cubed. Plus 6 X squad minus 4 X, and this will all be divided by X to the power of 4. Next, what we need to do is we need to distribute the -1 to the quantity 8X cubed plus 6X2 minus 4 X. That will allow us to simplify the derivative to be 8 X cubed plus 3 X2 minus 8 X cubed minus 6 X2 + 4 X. And again, this will all be divided by X to the power of 4. Now, what we will do is we will go ahead and combine like terms in the numerator. Combining 86 cubed with negative 8X cubed will reduce the terms to zero. We can combine 3x2 with -6X2 to a negative 3 x 2. So, we can rewrite the derivative to be 4 X minus 4X. I'm sorry, 3 x quantity squared. And this will all be divided by X to the power of 4. We can further simplify the derivative by rewriting this as a difference of fractions. So we can rewrite the derivative as 4 X divided by X to the power of 4 minus 3X 2 divided by X the power of 4. In the first fraction, we can reduce an X from both the numerator and denominator, and in the second fraction, we can reduce an X squared from both the numerator and denominator. That will allow us to rewrite the derivative as 4 divided by X cubed minus 3 divided by X2. There is no further way to simplify the derivative, so this will be the simplest form of the derivative of the given function. And with that being said, the solution to this problem is going to bed. So, I hope this video helps you in understanding how to use the quotient rule, and I will go ahead and see you all in the next video.

Derivatives Find and simplify the derivative of the following functions.
g(t) = t³+3t²+t / t³

Key Concepts

Derivatives

Quotient Rule

Simplification of Derivatives

Watch next