1. \(f(x) = 5\pi^2\\ f'(x) = 0\text{ as it is a constant}\)
2. \(f(x) = 5\sqrt x = 5x^{1/2}\\ f'(x) = \dfrac{5}{2}\cdot x^{-1/2}=\dfrac{5}{2\sqrt x}\)
3.\(f(x) = 1x^4 = x^4\\ f'(x) = 4 x^3\)
4.\(f(x) = 2x^7 - 5x - 7\\ f'(x) = 14x^6- 5\)
5.\(f(x) = 2x - 1x^3\\ f'(x) = 2 - 3x^2\\ f'(1) = 2- 3(1)^2 = -1\)
6. \(f(x) = 3x^4\ln x\\ f'(x) = 12x^3\ln x + 3x^3\\ f'(1) = 12(1)^3 \cdot \ln 1 + 3(1)^3 = 3\)
7.\(f(x) = 8\sqrt{x + 6x^{34}}\\ f'(x) = 8\left(\left(204x^{33}+1\right)\left(\dfrac{1}{2\sqrt{x+6x^{34}}}\right)\right)\\\quad\quad\;=4\left(\dfrac{204x^{33}+1}{\sqrt{x+6x^{34}}}\right)\)
8.\(y = 5x^2 - 3\sqrt x + 5x^4\\ \dfrac{\mathtt{dy}}{\mathtt{dx}}=10x - \dfrac{3}{2\sqrt x} + 20x^3\)
9.\(y = (6x^3 + 2)(5x - 3)\\ \dfrac{\mathtt{dy}}{\mathtt{dx}}=(18x^2)(5x - 3)+(6x^3 + 2)(5) = 120x^3 -54x^2+10\)
10. \(f(t) = (8t-3)(2t+5)t - 7\\ f'(t) = (8t-3)(2t+5)+2t(8t-3)+8t(2t+5)=48t^2 +68t - 15\\ f'(1) = 48(1)^2 + 68(1) - 15 = 101\)
Finally some fun calculus questions. I was so bored in those algebra 1 in school. LOL
PS: I am 14!! :D
~The smartest cookie in the world
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