What is derivative/differentiation (mathematically)?

In mathematics, the derivative is a way to show rate of change: that is, the amount by which a function is changing at one given point. For functions that act on the real numbers, it is the slope of the tangent line at a point on a graph. The derivative is often written using "dy over dx" (meaning the difference in y divided by the difference in x). The d's are not variable, and therefore cannot be cancelled out.

${\displaystyle {\frac {dy}{dx}}}$

The derivative of y with respect to x is defined as the change in y over the change in x, as the distance between ${\displaystyle x_{0}}$ and ${\displaystyle x_{1}}$ becomes infinitely small (infinitesimal). In mathematical terms,

${\displaystyle f'(a)=\lim _{h\to 0}{\frac {f(a+h)-f(a)}{h}}}$

That is, as the distance between the two x points (h) becomes closer to zero, the slope of the line between them comes closer to resembling a tangent line.

Power functions (e.g. ${\displaystyle x^{a}}$) behave differently than linear functions because their slope varies (because they have an exponent).

Power functions, in general, follow the rule that ${\displaystyle {\frac {d}{dx}}x^{a}=ax^{a-1}}$. That is, if we give a the number 6, then ${\displaystyle {\frac {d}{dx}}x^{6}=6x^{5}}$

Another possibly not so obvious example is the function ${\displaystyle f(x)={\frac {1}{x}}}$. This is essentially the same because 1/x can be simplified to use exponents:

${\displaystyle f(x)={\frac {1}{x}}=x^{-1}}$

${\displaystyle f'(x)=-1(x^{-2})}$

${\displaystyle f'(x)=-{\frac {1}{x^{2}}}}$

In addition, roots can be changed to use fractional exponents where their derivative can be found:

${\displaystyle f(x)={\sqrt[{3}]{x^{2}}}=x^{\frac {2}{3}}}$

${\displaystyle f'(x)={\frac {2}{3}}(x^{-{\frac {1}{3}}})}$