Hi there,

There is no forum for basic calculus, so I will have to post in the university-level forum. I have been trying to refresh my maths for the last few months (I had forgotten everything, right down to how to do long division.) and I feel as if I have been making good progress.

I have a question about a very simple differentiation exercise:

Using the formula

\(\displaystyle f'(x) = \displaystyle\lim_{\delta x \to 0} \frac{f(x + \delta x) -f(x)}{\delta x}\)

find the derivative of \(\displaystyle \frac{1}{x^2}\)

Here is my attempt at a solution:

\(\displaystyle f(x + \delta x) = \frac{1}{(x + \delta x)^2}\)

\(\displaystyle f(x + \delta x) - f(x) = \frac{1}{x^2 + 2x \delta x + (\delta x)^2} - \frac{1}{x^2}\)

\(\displaystyle = \frac{x^2 - (x^2 + 2x \delta x + (\delta x)^2)}{(x^2 + 2x \delta x + (\delta x)^2)x^2}\)

\(\displaystyle = \frac{\delta x(- \delta x -2x)}{x^4 + 2x^3 \delta x + x^2(\delta x)^2}\)

\(\displaystyle \frac{f(x + \delta x) - f(x)}{\delta x} = \frac{\delta x(- \delta x - 2x)}{x^4 + 2x^3 \delta x + x^2(\delta x)^2} \times \frac{1}{\delta x}\)

\(\displaystyle = \frac{-2x - \delta x}{x^4 + 2x^3 \delta x + x^2(\delta x)^2}\)

\(\displaystyle f'(x) = \displaystyle\lim_{\delta x \to 0} \frac{-2x - \delta x}{x^4 + 2x^3 \delta x + x^2(\delta x)^2}\)

\(\displaystyle = \frac{-2x}{x^4}\)

\(\displaystyle = \frac{-2}{x^3}\)

It would be great if someone would tell me where I am going wrong.

Regards,

Evanator

There is no forum for basic calculus, so I will have to post in the university-level forum. I have been trying to refresh my maths for the last few months (I had forgotten everything, right down to how to do long division.) and I feel as if I have been making good progress.

I have a question about a very simple differentiation exercise:

Using the formula

\(\displaystyle f'(x) = \displaystyle\lim_{\delta x \to 0} \frac{f(x + \delta x) -f(x)}{\delta x}\)

find the derivative of \(\displaystyle \frac{1}{x^2}\)

Here is my attempt at a solution:

\(\displaystyle f(x + \delta x) = \frac{1}{(x + \delta x)^2}\)

\(\displaystyle f(x + \delta x) - f(x) = \frac{1}{x^2 + 2x \delta x + (\delta x)^2} - \frac{1}{x^2}\)

\(\displaystyle = \frac{x^2 - (x^2 + 2x \delta x + (\delta x)^2)}{(x^2 + 2x \delta x + (\delta x)^2)x^2}\)

\(\displaystyle = \frac{\delta x(- \delta x -2x)}{x^4 + 2x^3 \delta x + x^2(\delta x)^2}\)

\(\displaystyle \frac{f(x + \delta x) - f(x)}{\delta x} = \frac{\delta x(- \delta x - 2x)}{x^4 + 2x^3 \delta x + x^2(\delta x)^2} \times \frac{1}{\delta x}\)

\(\displaystyle = \frac{-2x - \delta x}{x^4 + 2x^3 \delta x + x^2(\delta x)^2}\)

\(\displaystyle f'(x) = \displaystyle\lim_{\delta x \to 0} \frac{-2x - \delta x}{x^4 + 2x^3 \delta x + x^2(\delta x)^2}\)

\(\displaystyle = \frac{-2x}{x^4}\)

\(\displaystyle = \frac{-2}{x^3}\)

It would be great if someone would tell me where I am going wrong.

Regards,

Evanator

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