This is sometimes written as: If y = x2 +1 then y’ = 2x.

Exponentials

Consider the variable is an exponent, such as 

y = ax

Then following same procedures as above

To explore this I have used spreadsheets to create a chart of diminishing δx, initially for a2 and a3. This produced the following:

Using equation (3) 

For ease of typing I will let  h = 𝜹x

So

Note: Both 2x and 3x unchanged and dy/dx is a constant times the original exponential. For 2x the differential is below 2x and for 3x it is above. This prompts the question “is there a value for the base which results in the differential equaling the exponential?”

To explore this I repeated the spreadsheet using a = 2.75 and a = 2.71828. Now the second number is in fact the Euler constant e but if one was doing this exercise by considering all values then ultimately this number would be tried, but let’s treat it as a lucky guess.

Observe that for 

Note: As mentioned previously 2.71828 is an approximation of Euler’s constant e and the important result in calculus 

Consider a change of 𝜹x = h in x  results in a change of 𝜹y.

This leads to the most important result 

Note:  For convenience logex is written as ln x and log10x as log x.

Integration is dealt with in Integration but it is relevant to this section for the reasons shown below.

Integration can be considered to be the opposite of differentiation. Its symbol is ∫  and it asks the question what  function g(x) gives the function f(x) ?

Example