5–24. For each of the following composite functions, find an i...

Question

5–24. For each of the following composite functions, find an inner function u=g(x) and an outer function y=f(u) such that y=f(g(x)). Then calculate dy/dx.

y = (5x²+11x)^4/3

Accepted Answer

Hello there. Today we're gonna 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. Identify the inner function U equals G of X, and the outer function Y equals F of U for the composition Y equals F of G of X. Then find Dy by DX. Y equals 4 x 4 minus 9 x 2 + 1 all the power of 5/4. Awesome. So it appears for this particular prompt, we're asked to solve for three separate answers. We're asked to identify the inner function as our first answer. Our second answer, we're asked to identify the outer function, and thirdly, we're asked to solve for Dy by DX of our Y expression. Awesome. So with that said, our first step, let's identify the inner function, and when we do that, we will find that the inner function is U equals 4. Which is, it's technically 4 multiplied by X to the power of 4, but we could say 4 X to the power of 4 minus 9 X to the power of 2. Plus 1. Now we need to identify the outer function, which the outer function is Y equals F of U, which is equal to the power of 5 quarts. So now, our next step is we need to recall and use the chain rule. So we need to take DY by DX and we need to say that this is equal to Dy by DU, and we need to multiply this by DU by DX. So now we need to find DU by DX. So when we do that, we will find that DU, so this is DU by D. X is equal to D by DX. Of 4 X 4 minus 9 x 2 + 1. Which is equal to 4 multiplied by D by DX. Of X to the power of 4 minus 9 multiplied by DDX. Multiplied by X to the power 2. Plus, Drum roll. D by DX of 1, which this will mean that it's all equal to, let me. Perform the math, 16 X 3 minus 18 X. So now we need to find our next step now is we need to find Dy by DU and Dy by DU is equal to 5/4, so 5 4s. Multiplied by U to the power of 1/4. So now we need to substitute in our value of you back into our expression. So when we do that, we will find that Once again, that D by DX is equal to 5/4s multiplied by 4 X 4 minus 9 x 2 + 1, all to the power of 1/4. Multiplied by 16 X to the power of 3 minus 18 X. Fantastic. So now our next step is we need to factor completely. So once again, to reiterate, we need to factor completely and combine our like terms. So when we do that, we will find that DY by DX. Is going to be equal to. 5 X divided by 2 multiplied by 4 x 4 minus 9 x 2 plus 1 all the power of 1/4 multiplied by 8 x 2 minus 9. So that will mean that our final answer, and let's write it up above in green, that our final answer will be that you is equal to 4 X to the power of 4. Minus 9 X to the power of 2 + 1, and that F of you is going to be equal to, so F for you. is equal to the power of 5/4, and that Dy by DX is going to be equal to 5 x divided by 2. Multiplied by 4 x 4 minus 9 x 2 plus 1, all to the power of 1/4 all multiplied by 8 X to the power of 2 minus 9. Fantastic, we did it! Woo! So, let us quickly Right, or let's put a box around our final answers, and that's it. We've solved for all three answers we were asked to solve for. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

5–24. For each of the following composite functions, find an inner function u=g(x) and an outer function y=f(u) such that y=f(g(x)). Then calculate dy/dx.
y = (5x²+11x)^4/3

Key Concepts

Composite Functions

Chain Rule

Power Rule

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