MATH1131 Chap.8 Derivatives, Implicit Differentiation and Rates
Derivatives, Implicit Differentiation and Rates
The derivative is defined as the limit of the difference quotient, and it measures the slope of the tangent and the instantaneous rate of change. From that definition come the sum, product, quotient and chain rules, and the important fact that a differentiable function is continuous, while the converse fails at corners such as |x| at zero.
Piecewise functions are differentiable at a join only when they are continuous there and the one-sided derivatives agree. Implicit differentiation finds slopes on curves defined by equations in x and y, treating y as a function of x and applying the chain rule wherever y appears, so it handles circles, ellipses and other curves that are not graphs of single functions.
Related rates problems apply the same idea with respect to time: write an equation linking the changing quantities, differentiate it with respect to t, and substitute the instantaneous values only afterwards. The chapter also introduces the tangent-line approximation, local extrema, and stationary points where the derivative is zero, which prepare for the mean value theorem.
What this chapter covers
- 01
Derivative as a limit of difference quotients
- 02
Rules of differentiation and the chain rule
- 03
Differentiability implies continuity
- 04
Differentiability of piecewise functions
- 05
Implicit differentiation
- 06
Related rates
- 07
Linear approximation
- 08
Local extrema and stationary points
Worked example · free
Related rates for a growing circle
- 1Relationship: A = pi r^2, valid at every time.
- 1Differentiate with respect to t using the chain rule: dA/dt = 2 pi r dr/dt.
- 1Substitute r = 10 and dr/dt = 3 only now: dA/dt = 2 pi (10)(3) = 60 pi.
- 1Interpret: the area grows at 60 pi, about 188.5, square centimetres per second at that instant.
Key terms
- Derivative
- The limit of the difference quotient, giving the slope of the tangent at a point when it exists.
- Chain rule
- The rule that the derivative of a composite function is the outer derivative evaluated at the inner function times the inner derivative.
- Implicit differentiation
- Differentiating an equation in x and y term by term while treating y as a function of x.
- Stationary point
- A point where the derivative of a differentiable function equals zero.
Derivatives, Implicit Differentiation and Rates FAQ
Why is |x| not differentiable at zero?
The difference quotient approaches minus one from the left and one from the right, so the two-sided limit does not exist. The graph has a corner there even though it is continuous.
How do I differentiate xy implicitly?
Use the product rule, treating y as a function of x: the derivative is y plus x times dy/dx. Then collect every term containing dy/dx on one side and solve.
Why should I substitute values only after differentiating in related rates problems?
Substituting a current value first turns a changing quantity into a constant, so its rate disappears from the equation. Differentiate the relationship that holds at all times, then insert the instantaneous values.
What does it mean for a piecewise function to be differentiable at the join?
It must first be continuous there, and then the derivative from the left must equal the derivative from the right. If either condition fails, the derivative does not exist at that point.
How is a tangent line approximation used to estimate values?
Replace the function near a known point by its tangent line, f(a) plus f'(a) times x minus a, and evaluate that line instead. The estimate is accurate for x close to a and its error depends on the curvature.
Exam move
Practise differentiation rules until they are automatic, then spend most of your time on the setup of implicit and related rates problems, where marks depend on reasoning. For each related rates problem, write a short list of variables with units, the equation that links them, and which rates are given and wanted, before differentiating. For piecewise questions, always check continuity before comparing one-sided derivatives.
Use the linear approximation on a few numbers you can check with the online calculator to see how its error behaves. Draw and label a diagram for every related rates problem before writing any equation; most setup errors come from mixing up which length is changing and which is fixed.
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