Introduction to Differentiation

Definition 1:
To understand the concept of basic differentiation, we need to consider the gradient or slope or steepness of a straight line of the form y=mx+c where y and x are variables, m the gradient of the line and c is the y-intercept. Now the gradient of a straight line is the change in the vertical distance over the change in the horizontal distance. That is;
Gradient, m=(change in vertical distance)/(change in the horizontal distance)

Note that the gradient of a straight line is constant, but the gradient of a curve keeps on varying.

Definition 2:
To determine the gradient of a curve at point we need to use a gradient function. Given a function;
y=f(x)=kx^{n}
Then the gradient function;
dy/dx=knx^{n-1}

Example 1: Let, y=3x^{5},then dy/dx=15x^{4}

Example 2: Let; y=2 , then dy/dx =0

Remark 1: Note that, y=2 can also be written as y=2x^{0}. Recall that x^{0}=1 where x is not 0

Example 3: Differentiate the following function with respect to x; y=(2x-3)(4x+5)

Solution: Note that another term that is used to mean, 'finding the gradient function' is 'differentiate' with respect to a certain variable. One can also refer differentiation as 'determining the derived function'.
In this example, we need to expand the RHS to get;
y=8x^{2}-2x-15 then dy/dx =16x-2

Remark 1: The rule so far discussed is the most basic rule of differentiating. Some functions may not be differentiated at all over certain intervals or may not be differentiated using the above rule. Some differentiating techniques are discussed in the next section while others are beyond the scope of this blog.

P.S.
1) Next post will be on techniques of differentiating
2) This is an excerpt from the textbook, Basic Mathematics; by Kahenya NP