Mathematics Marathon (2 Viewers)

azureus88 · May 4, 2009

[maths]\int_{0}^{4}(4-x)dx\\=[4x-\frac{x^2}{2}]^4_0\\=8[/maths]

[maths]V=\pi\int_{0}^{4}(4-y)dy\\=\pi\int_{0}^{4}(4-x)dx\\=8\pi[/maths]

New Question:

Find dy/dx for:

[maths]y=\ln(\frac{x^{99}\sqrt[5]{5x+1}}{(x+1)^{21}\sqrt[3]{x+1}})[/maths]

Do not expand it out lol

Drongoski · May 4, 2009

Here's my attempt:

$\100dpi To\ show:\ \ \int e^{f(x)}dx\ \ = \ \ \frac{e^{f(x)}}{f'(x)}\ \ \Rightarrow \ \ f(x)\ \ is\ linear.\\ \\ \\ \\ \ \frac{\mathrm{d} }{\mathrm{d} x}\left (\frac{e^{f(x)}}{f'(x)} \right )\ \ =\ \ \frac{f'(x)\left [e^{f(x)}f'(x) \right ]\ -\ e^{f(x)}\cdot f''(x)}{\left [f'(x) \right ]^{2}}\ \ =\ \ e^{f(x)}\\ \\ \\ \therefore \ \ e^{f(x)}\left (\left [f'(x) \right ]^{2}\ -\ f''(x) \right )\ \ =\ \ e^{f(x)}\left [f'(x) \right ]^{2}\\ \\ \therefore \ \ f''(x)\ \ =\ \ 0\\ \\ \therefore \ \ f(x)\ is\ linear\\ \\ \\\left \{because,\ if\ reqd\ to \ show:\ \ f'(x)\ =\ c\ \ \ \therefore \ \ f(x)\ =\ cx\ +\ d \ \ [linear\ function] \right \}$

Drongoski · May 4, 2009

$\100dpi y\ \ =\ \ 98\ lnx\ +\ \frac{1}{5}\ ln(5x+1)\ -\ 21\ ln(x+1)\ -\ \frac{1}{3}\ ln(x+1) \\ \\ \therefore \ \ y'\ \ =\ \ \frac{98}{x}\ + \frac{1}{5x+1}\ -\ \frac{21}{x+1}\ -\ \frac{1}{3(x+1)}$

Have I expanded ??

Mathematics Marathon (2 Viewers)

azureus88

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Drongoski

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Drongoski

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