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- Derivatives
- The Derivative function
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The Derivative function
Given the function $f(x)=x^2+x$, compute the function derived from its definition.
According to the definition, $$\displaystyle f'(x)=\lim_{\Delta x \to 0}\frac{f(x+\Delta x)-f(x)}{\Delta x}$$ that is, $$\displaystyle f'(x)=\lim_{\Delta x \to 0}\frac{f(x+\Delta x)-f(x)}{\Delta x}=\lim_{\Delta x \to 0}\frac{(x+\Delta x)^2+(x+\Delta x)-(x^2+x)}{\Delta x}=$$ $$=\lim_{\Delta x \to 0}\dfrac{x^2+2x\Delta x+\Delta x^2+x+\Delta x-x^2-x}{\Delta x}=\lim_{\Delta x \to 0}(\Delta x+2x+1)=2x+1$$
$f'(x)=2x+1$