# 21-111, The derivative, Ch1

Yesterday I have managed to complete my next chapter of Brief Calculus & Its Applications.

The following topics were covered by Chapter 1:

• The Slope of a Straight Line
• The Slope of a Curve at a Point
• The Derivative and Limits
• Limits and the Derivative
• Differentiability and Continuity
• Some Rules for Differentiation
• The Derivative as a Rate of Change

So far so good but I guess my pace is to high right now. I will spend some time going through the book once again just in case I missed anything.

~450 pages left to finish. I might slow down a little bit but hope to be done with the book by the end of next week.

A little bit of theory. Lets start with The Slope of a Straight Line:

We can compute the slope of a line by knowing two points on the line. If (x1,y1) and (x2,y2) are on the line, the slope of the line (m) is:

`m = (y2 − y1) / (x2 −x1)`

The slope of a curve at a point P is defined to be the slope of the tangent line to the curve at P and follows the slope formula

` slope of the graph of y = x^2 at the point (x,y) = 2x`

The slope formula that gives the slope of the curve y = f(x) at any point is called the derivative of f(x) and is written f′(x). In other words – the derivative f′(a) measures the rate of change of f(x) at x = a.

There are 3 rules we need to remember:

1. If f(x) = mx + b, then we have f′(x) = m
2. The derivative of a constant function f(x) = b is zero. That is, f′(x) = 0
3. Power rule: let r be any number and let f(x) = x^r. Then f′(x) = rx^(r−1)

Examples of Power rule

1. If f (x) = x^2 , then its derivative is the function 2x. That is, f′(x) = 2x
2. If f (x) = x^3 , then the derivative is 3x^2 . That is, f′(x) = 3x^2
3. If f (x) = 1, then f′(x)=−1 (x̸0)

And now we can also write Equation of the Tangent Line

`y − f(a) = f′(a)(x − a)`

Another important thing to remember is that on the tangent line, the change in y, for one unit change in x, is equal to the slope f′(a)

`f(a + 1) − f(a) ≈ f′(a) OR f(a + 1) ≈ f(a) + f′(a) `

And to summarise everything