Matematica per le superiori/Derivate: differenze tra le versioni

Per la definizione di derivata:

:<math> \lim_{\Delta x \rightarrow 0}{\frac{(f(x + \Delta x) \cdot g(x + \Delta x)) - (f(x) \cdot g(x))}{\Delta x}} =</math>

 

Notando che: <math>+ f(x) \cdot g(x + \Delta x) - f(x) \cdot g(x + \Delta x) = 0</math>, il limite qui sopra si può scrivere come:

:<math> \lim_{\Delta x \rightarrow 0}{\frac{f(x + \Delta x) \cdot g(x + \Delta x) - f(x) \cdot g(x) + f(x) \cdot g(x + \Delta x) - f(x) \cdot g(x + \Delta x)}{\Delta x}} =</math>

:<math>\lim_{\Delta x \rightarrow 0}{\left\{ g(x + \Delta x) \cdot \left[ \frac{f(x + \Delta x) - f(x)}{\Delta x}\right] + f(x) \cdot \left[ \frac{g(x + \Delta x) - g(x)}{\Delta x} \right] \right\} } =</math>

:<math>\lim_{\Delta x \rightarrow 0}{\left\{ g(x + \Delta x) \cdot \left[ \frac{f(x + \Delta x) - f(x)}{\Delta x}\right] \right\} } + \lim_{\Delta x \rightarrow 0}{ \left\{ f(x) \cdot \left[ \frac{g(x + \Delta x) - g(x)}{\Delta x} \right] \right\} } =</math>

:<math>\lim_{\Delta x \rightarrow 0}{g(x + \Delta x)} \cdot \lim_{\Delta x \rightarrow 0}{\frac{f(x + \Delta x) - f(x)}{\Delta x}} + \lim_{\Delta x \rightarrow 0}{f(x)} \cdot \lim_{\Delta x \rightarrow 0}{ \frac{g(x + \Delta x) - g(x)}{\Delta x}} =</math>

:<math> = f'(x) \cdot g(x) + f(x) \cdot g'(x)</math>

:c.v.d

 

 

Per la definizione di derivata:

:<math> \lim_{\Delta x \rightarrow 0}{\frac{\left( \frac{f(x + \Delta x)}{g(x +\Delta x} \right) - \frac{f(x)}{g(x)}}{\Delta x}} = \lim_{\Delta x \rightarrow 0}{\frac{\frac{f(x + \Delta x) \cdot g(x) - f(x) \cdot g(x + \Delta x)}{g(x + \Delta x) \cdot g(x)}}{\Delta x}} =</math>

 

Notando che: <math>+ f(x) \cdot g(x) - f(x) \cdot g(x) = 0</math>, il limite qui sopra si può scrivere come:

:<math> \lim_{\Delta x \rightarrow 0}{\frac{1}{g(x + \Delta x) \cdot g(x)} \cdot \frac{f(x + \Delta x) \cdot g(x) - f(x) \cdot g(x + \Delta x) + f(x) \cdot g(x) - f(x) \cdot g(x)}{\Delta x}} =</math>

:<math>\lim_{\Delta x \rightarrow 0}{\frac{1}{g(x + \Delta x) \cdot g(x)}} \cdot \lim_{\Delta x \rightarrow 0}{\left\{ g(x) \cdot \left[ \frac{f(x + \Delta x) - f(x)}{\Delta x} \right] - f(x) \cdot \left[ \frac{g(x + \Delta x) - g(x)}{\Delta x} \right] \right\} } =</math>

:<math>\frac{1}{\left[ g(x) \right]^2} \cdot \left[ \lim_{\Delta x \rightarrow 0}{g(x)} \cdot \lim_{\Delta x \rightarrow 0}{\frac{f(x +\Delta x - f(x)}{\Delta x}} - \lim_{\Delta x \rightarrow 0}{f(x)} \cdot \lim_{\Delta x \rightarrow 0}{\frac{g(x + \Delta x) - g(x)}{\Delta x}} \right] =</math>

:<math> = \frac{f'(x) \cdot g(x) - g'(x) \cdot f(x)}{\left[ g(x) \right]^2}</math>

:c.v.d.

 

'''<math>y = x^n</math>

:<math>f'(x) = \lim_{\Delta x \rightarrow 0}{\frac{f(x + \Delta x) - f(x)}{\Delta x}} = \lim_{\Delta x \rightarrow 0}{\frac{(x + \Delta x)^n - x^n}{\Delta x}} =</math>

:<math>\lim_{\Delta x \rightarrow 0}{\frac{\left( x^n + n \cdot x^{n - 1} \cdot (\Delta x) + \binom{n}{2} \cdot x^{n - 2} \cdot (\Delta x)^2 + ... + n \cdot x \cdot (\Delta x)^{n - 1} + (\Delta x)^n \right) - x^n}{\Delta x}} =</math>

:<math> \lim_{\Delta x \rightarrow 0}{\frac{(\Delta x) \cdot \left( n \cdot x^{n - 1} + \binom{n}{2} \cdot x^{n - 2} \cdot (\Delta x) + ... + n \cdot x \cdot (\Delta x)^{n - 2} + (\Delta x)^{n - 1} \right)}{\Delta x}} =</math>

:<math>\lim_{\Delta x \rightarrow 0}{n \cdot x^{n - 1} + (\Delta x) \cdot \left[ \binom{n}{2} \cdot x^{n - 2} + ... + n \cdot x \cdot (\Delta x)^{n - 3} + (\Delta x)^{n - 2} \right] } = </math>

:<math> n \cdot x^{n - 1} \Rightarrow y' = n \cdot x^{n - 1}</math>

 

 

[[Categoria:Matematica per le superiori|Derivate]]