In the following applet, you can explore how this process works. This definition of derivative of f (x) is called the First Principle of Derivatives. Isaac Newton and Gottfried Wilhelm Leibniz independently developed the theory of infinitesimal calculus in the later 17th century. We are using the example from the previous page (Slope of a Tangent), y = x2, and finding the slope at the point P(2, 4). Differentiating a linear function A straight line has a constant gradient, or in other words, the rate of change of y with respect to x is a constant. Use the left-hand slider to move the point P closer to Q. CALCULUS First Principles = 3 lim h!0 (x+ h) x h lim h!0 (x+ h)4 x4 h I Identify the fundamental limits: = 3L 1 L 4 I Substitute the values of the fundamental limits: = 3(1) (4x3) I Simplify: = 3 4x3 8. f0(x) = lim h!0 f(x+ h) f(x) h I Use substitution: = lim h!0 [ 2(x+ h) + 2(x+ h)4] [ 2x+ 2x4] h I Group the like terms and factor: = 2 lim h!0 (x+ h) x h + 2 lim h!0 Consider the straight line y = 3x + 2 shown below. Before calculus was developed in the 17th century, the only way to find the slopes, areas under a curve and curve lengths was to draw rectangles or trapezoids with increasingly smaller widths to get a good approximation. The process of finding the gradient value of a function at any point on the curve is called differentiation, and the gradient function is called the derivative of f (x). W… Calculus, known in its early history as infinitesimal calculus, is a mathematical discipline focused on limits, continuity, derivatives, integrals, and infinite series. Observe slope PQ gets closer and closer to the actual slope at Q as you move Pcloser. You can get an idea how this works in the following applet. Example. A graph of the straight line y … f' (x) =. This section looks at calculus and differentiation from first principles. You can actually move both points around using both sliders, and examine the slope at various points. The function f' (x) or is called the gradient function.

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