Given that p(x) = (5e^x+10x⁵+20x³+100x²+5x+20) ⋅ (10x⁵+40x³+20...

Question

Given that p(x) = (5e^x+10x⁵+20x³+100x²+5x+20) ⋅ (10x⁵+40x³+20x²+4x+10), find p′(0) without computing p′(x).

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. For the following function, determine R of 0 without computing its derivative. R of X is equal to 7 multiplied by e to the power of X plus 10 x 8 plus 10 x 7 + 200 X. 3 plus 29 x plus 50 multiplied by 25 X to the power of 8 plus 20 x 6 plus 100 X to the power of 4 plus 50 X plus 25. Awesome. So it appears for this particular problem we're trying to determine our of zero without computing its derivative. So now that we know that we're trying to determine our of 0, let's read off our multiple choice answers to see what our final answer might be. A is 3750, B is 2850. C is 900 and D is 1950. Awesome So first off, let us state that P X is equal to 7 multiplied by E of X plus 10 8 + 10 7 + 200. X to the power of 3 + 29 X plus 50, and let us say that Q of X. Is Equal to 25 x 8 plus 20 x 6 + 100 x 4 plus 50 plus 50 X plus 25. And if you're wondering why are we saying like what is PF X and what is Q of X? Well, these are two separate functions, so we're taking our big old long function for RFX and we're splitting it into two separate functions because if you haven't noticed, we're gonna have to recall and use the quotient rule to help us solve for this particular problem. And because we're using the quotient rule, that's why we have to separate our long original function for RFX and split it into two separate functions. So now, let us state that R of X is equal to PFX multiplied by Q X. So we need to know and recall that the derivative of R of X is given as R of X is equal to P X. So, like, once again, so the prime symbol means the derivative, so the derivative of R of X, so this is R of X, so R of X is equal to PX multiplied by Q X plus PX multiplied by Q X. So in order to find the derivative of PF X and Q of X, We need to note and recall that P X is equal to 77 multiplied by the power of X plus 80X to the power of 7 plus 70. X to the power of 6 + 600 X to the power of 2, plus 29. And we need to note that Q X is equal to 200 X 7 + 120 x 5 plus 400 X 3 + 50. Fantastic. So now we need to find the value of PF X, Q of X, X, and Q X when X equals 0, so noting that this is when X equals 0. Awesome. So we could say that P of 0, so we're gonna say P of 0. is equal to 7 multiplied by the power of 0 plus 10 multiplied by 0 to the power of 8 plus 10 multiplied by 07 + 200 multiplied by 0 to the power of 3 plus 29 multiplied by 0, plus 50 is gonna be all equal to 57. Fantastic. So now similarly for Q of 0, we could say that Q of 0 is equal to 25 multiplied by 0 to the power of 8 plus 20 multiplied by 06 + 100 multiplied by 0 of 4 plus 50 multiplied by 0, plus 25 is all equal to 25. Fantastic. So now, our next step is we need to determine P 0 and P 0 is equal to 7 multiplied by E to the power of 0. Plus 80 multiplied by 0 to the power of 7. Awesome, moving right along here. So now we need to take 80 multiplied by 0 to the power of 7, and we need to add 70 multiplied by 0 to the power of 6, plus 600 multiplied by 0 to the power of 2, plus 29. Which is going to be all equal to when we plug in, uh, when we plug in this into our calculator, we will find that it's equal to 36, and similarly solving for Q. So Q 0 is going to be equal to 200 multiplied by 0 to the power of 7 plus 120 multiplied by 0 to the power of 5 plus 400, so this is gonna be plus 400. Multiplied by 0 to the power of 3 plus 50, which will be all equal to 50 when we plug that into our calculator. So now our next step is we need to substitute in the required values into our R X equation. So, once again, as we should recall, R of X is equal to P. Of X multiplied by Q X plus PX multiplied by Q X. So when we plug in our known variables, we will find that R of 0 is equal to P0 multiplied by Q 0 plus P 0 multiplied by Q 0. Which is equal to 36 multiplied by 25 plus 57 multiplied by 50, so 36 multiplied by 25 plus 57 multiplied by 50 will be equal to, when you plug that into your calculator, 3750. And that is it. That is our final answer. Hooray, we did it. So, going back to look at our multiple choice answers, the correct answer, without a doubt has to be multiple choice answer letter A. Which once again is 3750. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

Given that p(x) = (5e^x+10x⁵+20x³+100x²+5x+20) ⋅ (10x⁵+40x³+20x²+4x+10), find p′(0) without computing p′(x).

Key Concepts

Product Rule

Evaluating Derivatives at a Point

Exponential and Polynomial Functions

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