We are familiar with finding the gradient of a straight line - we use [size=200][color=#666666]two points[/color][/size] that lie on the line and divide their difference in [i]y[/i] by their difference in [i]x:[br][/i][br][math]\frac{\Delta y}{\Delta x}=\frac{y_2-y_1}{x_2-x_1}[/math][br][br]What about finding the gradient of a curve then? Well, this is a little more tricky. [br][br]Firstly we need to be familiar with the notion of a [size=200][color=#666666]tangent [/color][/size]- it's a line which [i][size=150][color=#999999]just touches[/color][/size][/i]the curve at one point. [br][br]As you can see in the activity below, as we [size=150][b][color=#666666]zoom in[/color][/b][/size] more and more, the curve approaches a straight line - in fact it approaches the gradient of the tangent. If we were to keep zooming in an infinite amount of times, eventually the curve would become exactly a straight line. When will this happen? Well when we are looking at an [size=200][color=#666666]infinitesimal [/color][/size]change - that is a change so small that if it were any smaller it would be zero (think of it as the opposite of infinity).[br][br]The idea of an infinitesimal lies at the heart of calculus and is what makes it work [br][br]However finding the gradient of this tangent is a problem for two reasons:[br][br][list=1][*]The gradient will be different at different points (it's a curve!)[/*][*]In order to compute gradient, we need [size=200][color=#666666]two [/color][/size]points - a tangent has just [size=200][color=#666666]one [/color][/size](by definition - it [i][size=150][color=#999999]just touches[/color][/size][/i] the curve)[/*][/list]So the solution is to find a point [size=150][color=#999999][/color][/size][size=150][color=#999999]really close-by[/color] [/size]and find the gradient of the line that passes through these two points (by the way - this line is called a [b][size=150]secant[/size][/b]). Then all we have to do is move these points [size=200][color=#999999]closer and closer together[/color][/size] - notice on the activity below that the gradient of the secant will approach the gradient of the tangent. Then, when the distance between the two points is [size=200][color=#666666]infinitesimal[/color][/size], (we call this idea a [b][size=150]limit[/size][/b]) the gradient of the secant must equal the gradient of the tangent![br][br]And that's how we find the gradient of a curve!

Below is a derivation of the formula for Differentiation from First Principles. It's really just using our formula for gradient:[br][math]\frac{\Delta y}{\Delta x}=\frac{y_2-y_1}{x_2-x_1}[/math]