Derivatives

The derivative is, before any formula, a geometric concept: the slope of the tangent line to the graph. The chapter follows a three-step path. Step A: the derivative as the slope of the tangent. Step B: its formalisation as the limit of the difference quotient, with Leibniz’s notation $\frac{df}{dx}$ and the differential. Step C: the rules of differentiation, with the “circle diagram” method for the composite function. From here the chapter continues with the derivative of the inverse, the points of non-differentiability (corner, cusp, vertical tangent), the Taylor expansion with the three remainders, Fermat’s theorem and optimisation problems.

Sections

• The derivative as the slope of the tangent
• The derivative as the limit of the difference quotient
• Derivatives of the elementary functions
• Derivative of the composite function
• Derivative of the inverse function
• Points of non-differentiability
• Taylor expansion
• Fermat’s theorem
• Optimisation problems

Exercises

• All the worked exercises of the chapter