Simple calculus

When we are first introduced to Calculus we are usually required to just accept that if y = xn, then for any given value of x the rate of change of y with respect to x will be dy/dx = n xn-1

Lets see if we can make sense of why that is so by looking at the curve y = x3. Inset is a small section of the curve with a lowest x value of a-b and an upper x value of a + b.

What are the corresponding y values. They are (a-b)3 and (a+b)3 and similarly for any curve y = xn they would be (a-b)n and (a+b)n so let us look at expanding these terms.

The binomial theorum applies but we will use Pascal’s triangle. It tells us that (a+b)5 for instance is
a5 + 5a4b +10a3b2 +10a2b3 + 5ab4 +b5
Do you see the pattern – decending powers of a and ascending powers of b and numbers taken from the appropriate row of the triangle, the second number of which is always that of the power (n).

Can you see that (a-b)5 will equal a5 – 5a4b +10a3b2 -10a2b3 + 5ab4 -b5 and that (a-b)n will similarly alternate in sign because every odd power of a minus operation is a minus and every even power of a minus operation is a plus operation.

Let us now return to the slope of our graph and any graph y = xn . The vertical change is (a+b)n(a-b)n and looking at our patterns can you see that in the subtraction only the terms with odd powers of b will survive and that the number values will double.
For example (a+b)5(a-b)5 = 10a4b + 20a2b3 + 2 b5

If b is tiny then higher odd powers of b like b3 and b5 will be miniscule. By ignoring these the vertical (dy) change (a+b)n(a-b)n equals 2nan-1 b.
As our horizontal (dx) change is 2b the slope of the graph as at x = a will be 2nan-1 b / 2b = nan-1 i . e. for y = xn dy/dx = n xn-1

When in calculus we say let the change in x that is dx approach zero we aren’t making it zero and indeed its change is most relevant because without it there is no change and no slope on our graph.

The calculus formula that says when y = xn , that the change of y with respect to x will be n xn-1 works because if dx is tiny we can ignore the (dx)3 , (dx)5 and (dx)7, etc content in dy. At no stage do we ignore dx and in higher and more complex maths we have to be careful so as to not treat it as zero.