When you differentiate a function or curve you are finding the gradient of the tangent to the curve. If the curve you are differentiating a curve or function that is in terms of x, eg y=x², then the differential is also a curve or function in terms of x, eg 2x. This allows you to find the gradient of the curve for any point on the curve with a known x co-ordinate.

Differentiating from first principles involves finding the gradient of a tangent to a curve from the basic definition of gradient,
grad = (y₁ – y₂) / (x₁-x₂)
ie) not by following a rule.

When differentiating from first principles we consider chords (lines passing from one point on the curve to another) from the point on the curve with the x coordinate x to points with a slightly larger x coordinate x+d. The gradient of these chords is an approximation of the gradient of the tangent to the curve at the point with x co-ordinate x and the approximation becomes better as d becomes smaller. If we take the limit as d tends to 0 we find the actual gradient of the tangent

Note: the derivative of a curve y = f(x) is written as dy/dx

Example: y=x²

Consider 2 points P and Q on the curve y=x² a small distance apart where the difference in their x co-ordinate is d. Then
P(x,x²)
Q(x+d,(x+d)²)
Then the gradient of the chord PQ is given by

To find the derivative we must now take the limit of this as d tends to 0.
So we find

By David Woodford