# Product rule in differentiation

1. May 8, 2015

### Muthumanimaran

What has done here in the second line of the proof for product rule?, from Mathematical methods for physicists from Riley, Hobson
they defined f(x)=u(x)v(x) and these steps are given,

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2. May 8, 2015

### Simon Bridge

Do you know why they bothered to work out $f(x+\Delta x) - f(x)$ at all?
I mean: what's the point?

How would you normally go about proving the product rule - if you didn't have the example from Riley and Hobson?
i.e. do you know the definition of the derivative?

3. May 9, 2015

### Muthumanimaran

I don't know

yes, derivative is the rate of one function to another function, it actually says how fast one function changes with respect to other, am I right?

4. May 9, 2015

### Simon Bridge

Not exactly ... that was the description of what the derivative is, not the definition. The definition is: $$f^\prime(x) = \lim_{\Delta x\to 0}\frac{f(x+\Delta x) - f(x)}{\Delta x}$$ ... this gives the derivative of f(x) with respect to x.

To prove the product rule, first set $f(x)=v(x)u(x)$ then apply the definition to f.

5. May 9, 2015

### Muthumanimaran

got it. Thank you