Summary

This chapter developed the techniques of differentiation and applied them to business and geometric problems. Its key ideas and formulae are collected below.

Differentiation techniques

1. An implicit function is one in which the dependent variable is not expressed explicitly in terms of the independent variable; it is differentiated by differentiating both sides with respect to $x$ (treating $y$ as a function of $x$) and then solving for $\dfrac{dy}{dx}$.
2. For a parametric function $y=f(t),\ x=g(t)$:

3. Logarithmic differentiation is applied to functions of the type $f(x)^{g(x)}$ (such as $x^x$): take the logarithm of both sides first, then differentiate.
4. Second and higher order derivatives: the second-order derivative is $\dfrac{d}{dx}\!\left(\dfrac{dy}{dx}\right)=\dfrac{d^2y}{dx^2}=y''=f''(x)$, and the third-order derivative is $\dfrac{d}{dx}\!\left(\dfrac{d^2y}{dx^2}\right)=\dfrac{d^3y}{dx^3}=y'''=f'''(x)$.

Business applications

5. Cost function: $C(x)=V(x)+k$, where $V(x)$ is the variable cost and $k$ is the fixed cost.
6. Revenue function: $R(x)=p\cdot x$, where $p$ is the price per unit and $x$ is the output (sales) at price $p$.
7. $\dfrac{dy}{dx}$ is the rate of change (instantaneous rate) of $y$ with respect to $x$ — the change in $y$ for a very small change in $x$.
8. Marginal cost and marginal revenue are the derivatives of the cost and revenue functions:

Tangents and normals

9. The slope of the tangent to a curve at a point $A(x_0,y_0)$ is $\dfrac{dy}{dx}\Big]_{A(x_0,y_0)}$.
10. The slope of the normal at $A(x_0,y_0)$ is $\dfrac{-1}{\dfrac{dy}{dx}\big]_{A(x_0,y_0)}}$.
11. The equation of the tangent to the curve at $A(x_0,y_0)$ is $(y-y_0)=\dfrac{dy}{dx}\Big]_{(x_0,y_0)}(x-x_0)$.
12. The equation of the normal to the curve at $A(x_0,y_0)$ is $(y-y_0)=\dfrac{-1}{\dfrac{dy}{dx}\big]_{(x_0,y_0)}}(x-x_0)$.

Increasing, decreasing and monotonic functions

13. A real function $y=f(x)$ is an increasing function on $(a,b)$ if $f'(x)>0$ for all $x\in(a,b)$.
14. A real function $y=f(x)$ is a decreasing function on $(a,b)$ if $f'(x)<0$ for all $x\in(a,b)$.
15. A function that is either increasing or decreasing on its domain is termed a monotonic function.
16. An interior point $x_0$ of the domain is a critical point if either $f'(x_0)=0$ or $f$ is not differentiable at $x_0$.

Maxima and minima

17. If $f$ is defined on a domain $D$, then $f(c)$ is the absolute minimum value if $f(x)\ge f(c)$ for all $x\in D$, and the absolute maximum value if $f(x)\le f(c)$ for all $x\in D$; such a point $c$ is a point of extremum.
18. First derivative test at a critical point $c$: if $f'$ changes sign from negative to positive as $x$ increases through $c$, then $c$ is a point of local minimum; if it changes from positive to negative, $c$ is a point of local maximum; if $f'$ does not change sign, $c$ is a point of inflexion. …