The derivative: definition, differentiability and basic rules

Learning objectives

Define the derivative at a point as the limit of a difference quotient in either the h form or the x → a form, and recognize a given limit expression as the derivative of a particular function at a particular point

Contrast average rate of change, the slope of a secant line, with instantaneous rate of change, the slope of a tangent line, and estimate a derivative from a table or from the shape of a graph

Explain that differentiability at a point forces continuity there but that continuity does not force differentiability, and identify corners, cusps, vertical tangents and discontinuities as the places where a derivative fails to exist

Test a piecewise function for differentiability at a junction by matching both the function values and the one-sided derivatives, and solve for the parameters that make it differentiable

Apply the power rule for any real exponent together with the constant multiple, sum and difference rules, after rewriting radicals and reciprocals as powers, and quote the derivatives of sin x, cos x, eˣ and ln x

Write the equation of a tangent line at a point on a curve, and find every point at which the tangent has a specified slope