Maths with David

    IBDP. AA. HL. Differentiation. First Principles

    Combining what we’ve already cosidered, we know that

    If a limit exists, then f'(a) = lim_{ h \to 0 } \frac{ f(a+h) - f(a) }{h} is the gradient of the tangent to y = f(x) at x = a. We say that f(x) is differentiable at x = a.

    This definition lets us find the gradient of the tangent to a curve at a general point (x, f(x)) and hence find its derivative function.

    As per the diagram, the chord AB has gradient \frac{f(x+h) - f(x) }{ h } and as B approaches A, the chord AB becomes the tangent at A, so the gradient of the tangent at the general point (x, f(x) is lim_{h \to 0 } \frac{f(x+h) - f(x)}{h} . This formula is a function, because there is at most one value of the gradient for each value of x.

    The derivative function (the “derivative”) of f(x) is defined as f'(x) = lim_{ h \to 0 } \frac{f(x+h) - f(x)}{h} . The domain of f'(x) is the set of values for which the limit exists.

    When we use this strict limit definition to find a derivative function, we say that we are differentiating from first principles.

    Worked Example 1

    Use first principles to find the gradient function f'(x) of f(x) = x4 and then find f'(-1) and interpret your answer.

    Worked Example 2

    Prove that lim_{ h \to 0 } \frac{cos(h) - 1}{h} = 0 and hence find the derivative of f(x) = sin x.

    Exercise

    Answers