How dn/dxn becomes dn-1/dxn-1 …...please explain

Hello student,Please find answer to your questionFor example,Apply this

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Differential Calculus> How d n /dx n becomes d n-1 /dx n-1 …...p...

How dⁿ/dxⁿbecomes d^n-1/dx^{n-1 …...please explain}

milind , 10 Years ago

Grade 12

2 Answers

Jitender Singh

Last Activity: 10 Years ago

Hello student,
Please find answer to your question
$y = f(x)$
$\frac{d^{n}}{dx^{n}}(f(x)) = \frac{d^{n-1}}{dx^{n-1}}(f'(x)) = \frac{d^{n-2}}{dx^{n-2}}(f''(x)) = ...........$
For example,
$f(x) = (x+1)^{4}$
$\frac{d^{4}}{dx^{4}}(x+1)^{4} = 4!$
Apply this
$\frac{d^{n}}{dx^{n}}(f(x)) = \frac{d^{n-1}}{dx^{n-1}}(f'(x)) = \frac{d^{n-2}}{dx^{n-2}}(f''(x)) = ...........$
$\frac{d^{4}}{dx^{4}}(x+1)^{4} = \frac{d^{3}}{dx^{3}}(4(x+1)^{3}) = \frac{d^{2}}{dx^{2}}(4.3(x+1)^{2}) = \frac{d}{dx}(4.3.2.(x+1)) = 4!$

$f(x) = x^{n}log(x)$
$\frac{d^{n}}{dx^{n}}(f(x)) = \frac{d^{n-1}}{dx^{n-1}}(f'(x))$
$I_{n} = \frac{d^{n-1}}{dx^{n-1}}(x^{n}log(x))'$
$I_{n} = \frac{d^{n-1}}{dx^{n-1}}(x^{n}.\frac{1}{x}+log(x).nx^{n-1})$
$I_{n} = \frac{d^{n-1}}{dx^{n-1}}(x^{n-1}+nx^{n-1}log(x))$