Differentiating some functions from first principle

In this post, I am going to be describing how we find the derivative of some common functions, assuming no prior knowledge of the power rule, quotient rule or the product rule. So, without any further ado, let us differentiate!

i. x²+2x

Let us state the definition of a derivative:

Then, let us find the change in y (or delta-y):

Plugging delta-y into the formula above,

ii. x³+x²+x

Firstly,

Finding delta-y,

Plugging delta-y in the formula above,

iii. sin x

Finding delta-y,

Plugging it into the formula,

Now, this result might seem counter-intuitive, but, the limit of sin Δx/Δx as it tends to zero is 1 and the limit of (1-cosΔx)/Δx is 0.

That’s all for this article. I will soon be doing an article on the power rule, quotient rule, product rule and the chain rule.