+11 Question: Find the derivative for given functions.
1. π(π)=ππ
π
2. π(π)=πβπ
3. π(π)=πππ
4. π(π)=πππβππβπ
5. π(π)=ππβπππ , then find πβ²(π)
6. Let π(π)=ππππ₯π§π , then find πβ²(π)
7. If π(π)=πβπ+ππππβ, find πβ²(π)
8. If π=πππβπβπ+πππ, find π
ππ
π
9. If π=(πππ+π)(ππβπ), find π
ππ
π
10. π°π π(π)=(ππβπ)(ππ+π)πβπ ,then find πβ²(π).
+9665 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
+118658 Hi Max,
I know you are the smartest cookie in the world.
Everyone here knows it too.
So please do not do all of anyone's homework.
How much do you think someone learns when you do all their homework for them ?
Also,
When you post an answer can you be sure to include some ordinary type in it somewhere.
If it only has LaTex then other people cannot hyperlink into it from the answer page.
I usually get around this by copying the original question in first.
If the question is all latex then I would make up some thing to put first. :)
I am not referring to this answer, I am just talking about answers in general :)
Thanks :)
Max: You have to qualify your statement as "The smartest cookie IN CALCULUS!"
+9665 \(1.\\ f(x) = 5\pi^2\\ f'(x) = 0\text{ as it is a constant}\\ ......................................................\\ 2.\\ f(x) = 5\sqrt x\\ f'(x) = \dfrac{5}{2\sqrt x}\\......................................................\\ 3.\\ f(x) = \dfrac{1}{x^4}=x^{-4}\\ f'(x) = -4x^{-5}=-\dfrac{4}{x^5}\\......................................................\\ 4.\\ f(x) = 2x^7 - 5x^{-7}\\ f'(x) = 14x^6 +35x^{-6}=14x^6 + \dfrac{35}{x^6}\\......................................................\\ 5.\\ f(x) = \dfrac{2x-1}{x^3}=\left(x^{-3}\right)(2x-1)=2x^{-2}-x^{-3}\\ f'(x) = -4x^{-3} +3x^{-4}=-\dfrac{4}{x^3}+\dfrac{3}{x^4}\\ f'(1) = -\dfrac{4}{1^3}+\dfrac{3}{1^4}=-4 + 3 = -1 \\\text{P.S.: Need not use quotient rule nor product rule}\\ ......................................................\\ 6.\\ f(x) = 3x^4 \ln x\\ f'(x) = (12x^3)(\ln x)+(3x^4)(\dfrac{1}{x})=12x^3\ln x + 3x^3\\ f'(1) = 12(1^3)\ln 1 + 3(1^3)=12(0) + 3(1) = 3 \\......................................................\\ 7.\\ f(x)=8\sqrt x + 6x^{3/4}=8x^{1/2}+6x^{3/4}\\ f'(x) = \dfrac{4}{\sqrt x}+\dfrac{9}{2x^{1/4}}\\......................................................\\8.\\ y=\dfrac{5}{x^2}-\dfrac{3}{\sqrt x}+5x^4 = 5x^{-2} - 3x^{-1/2} + 5x^4\\ \dfrac{dy}{dx} = -10x^{-3} +\dfrac{3}{2x^{3/2}}+20x^3 = -\dfrac{10}{x^3}+\dfrac{3}{2x^{3/2}}+20x^3\\β......................................................\\ 9.\\ y = (6x^3 + 2)(5x - 3)\\ \dfrac{dy}{dx} = (18x^2)(5x - 3) + (6x^3 + 2)(5) = 120 x^3 -54x^2+10\\β......................................................\\ 10.\\ f(t) = \dfrac{(8t-3)(2t+5)}{t-7}= \dfrac{16t^2 +34t-15}{t-7} = 16t + 146 +\dfrac{1007}{t-7}\\ f'(t) = 16 - 1007(t-7)^{-2}=16-\dfrac{1007}{(t-7)^2}\\ f'(1) = 16 - \dfrac{1007}{(1-7)^2}=16 - 27\dfrac{35}{36} = -\dfrac{431}{36}\)
These questions correct
These questions correct
