A calculus fallacy

We all are fimiler with the following well known results in calculus:-

(1) d(xn)dx=nxn1\frac{d(x^n)}{dx}=nx^{n-1}

(2) ddx(f1(x)+f2(x))=f1(x)+f2(x)\frac{d}{dx}(f_1(x)+f_2(x))=f'_1(x)+f'_2(x)

Lets try to find the derivative of x2x^2 by using property (2):-

x2=x+x+x+...........+xx timesx^2=\underbrace{x+x+x+...........+x}_\text{x times}

Differentiating both sides of above equation:-

d(x2)dx=1+1+1+......+1x times\frac{d(x^2)}{dx}=\underbrace{1+1+1+......+1}_\text{x times}

d(x2)dx=x\frac{d(x^2)}{dx}=x

Which is horribly wrong,because d(x2)dx=2x\frac{d(x^2)}{dx}=2x

I can't identify the fallacy in it ,please help me

Note by Aman Sharma
4 years, 9 months ago

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1 vote

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A function ff is not differentiable if it is not continuous. While your reasoning is correct for xNx \in \mathbb{N}, it does not apply to xRNx \in \mathbb{R} \setminus \mathbb{N}; hence, ff is not continuous and therefore not differentiable.

Jake Lai - 4 years, 9 months ago

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Try writing π2\pi ^2 as π\pi a π\pi number of times. Or more simply 3.52 3.5^2 as a sum of 3.5 3.5 3.5 \text{ } 3.5 ' s . To make it more obvious just write 2525 as 5 5 added 5 5 times.

Sudeep Salgia - 4 years, 9 months ago

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When you are defining x^2=x+x+..+x(x times), you are tacitly defining the domain of f(x) to be N, the set of natural numbers. However, f(x) cannot be differentiable in the domain of N. This is because, if a function f defined from a domain D (a subset of R, the set of reals) to R (the set of reals) has to be differentiable at a real point c, then a necessary criterion is that c has to be a limit point of D and c has to be an element of D itself. In other words, c has to be a member of D such that every arbitrarily small neighbourhood of c has an element of D other than c. But no natural number is a cluster point of the set N. Here lies the fallacy. (last modified 1 hour ago )

Kuldeep Guha Mazumder - 4 years, 3 months ago

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very nicely explained. Thanks a ton mate.

Aman Sharma - 4 years, 2 months ago

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Happy to help you..:-)

Kuldeep Guha Mazumder - 4 years, 2 months ago

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In differentiation, we are concerned with the ratio of change of one quantity at a particular instant with respect to another.

See this lovely note by the great @Agnishom Chattopadhyay

@Aman Sharma

U Z - 4 years, 9 months ago

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