find the derivative of y with respect to x, t or theta. (a) e^7 - 10x Answer: -10e^7-10x (b) 8xe^x - 8e^x answer: 8xe^x (c) y= (x^2 -2x+4)e^x answer: (x^2+2)e^x (d) y= sin e^theta^4 answer: (-4theta^3 e^-theta^4) cos e^-theta^4 please show the steps. thank you

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find the derivative of y with respect to x, t or theta. (a) e^7 - 10x Answer: -10e^7-10x (b) 8xe^x - 8e^x answer: 8xe^x (c) y= (x^2 -2x+4)e^x answer: (x^2+2)e^x (d) y= sin e^theta^4 answer: (-4theta^3 e^-theta^4) cos e^-theta^4 please show the steps. thank you

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Ok It took me a while to understand your questions so here's how I saw them: a. \[f(x)=e^{7-10x}\] \[f'(x) = e ^{7-10x}\times \frac{ d(7-10x) }{ dx }\] (Chain Rule) Derivative of 7 - 10x = -10 so it comes out to the desired answer. b. Ok for this, I just took the 8 out, because it makes it much easier to look at. \[y=xe^{x}-e^{x}\] Derivative of e^x is simply e^x. Derivative of \[xe^{x}\] = \[e^{x}+xe^{x}\] (Product Rule) the e^x's cancel out and we are left with the desired answer.
C. \[f(x) = (x^{2}-2x+4)e^{x}\] Uhh this is also product rule so you get: \[(2x-2)e^{x}+e^{x}(x^{2}-2x+4)\] Expanding this out you get:\[2xe^{x}-2e^{x} + e^{x}x^{2}-e^{x}2x + 4e^{x}\] Terms cancel and you are then left with the desired answer.

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