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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:

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

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

Photoelectric Effect

The photoelectric effect is a phenomenon in physics. 

The effect is based on the idea that electromagnetic radiation is made of a series of particles called photons. When a photon hits an electron on a metal surface, the electron can be emitted. The emitted electrons are called photoelectrons. The effect is also called the Hertz Effect, because it was discovered by Heinrich Rudolf Hertz, but this name is not used often.

The photoelectric effect has helped physicistsunderstand the quantum nature of light and electrons. The concept of wave–particle duality was developed because of the photoelectric effect. Albert Einstein proposed the Laws of Photoelectric Effect and Won the Nobel Prize For Physics , 1921.

Not every electromagnetic wave will cause the photoelectric effect, only radiation of a certain frequency or higher will cause the effect. The minimum frequency needed is called the "cutoff frequency" or "threshold frequency'. The cutoff frequency is used to find the work function, w, which is the amount of energy holding the electron to the metal surface. The work function is a property of the metal and is not affected by the incoming radiation. If a frequency of light strikes the metal surface that is greater than the cutoff frequency, then the emitted electron will have some kinetic energy.

The energy of a photon causing the photoelectric effect is found through E = hf = KE + w, where h is Planck's constant, 6.626X10^(-34) J*s, f is the frequency of the electromagnetic wave, KE is the kinetic energy of the photoelectron and w is the work function for the metal. If the photon has a lot of energy, (~thousands of eV) or pair  of eV) may take place.

The intensity of the light does not cause ejection of electrons, only light of the cut off frequency or higher can do that. However increasing the intensity of light will increase the number of electrons being emitted, as long as the frequency is above the cut off frequency.

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