MTH6131 Chap.5 Rules of Differentiation and Implicit Differentiation
Rules of Differentiation and Implicit Differentiation
Rules proved once, used everywhere
First principles work for any function, but the lectures note that for composite functions they can be a fruitless endeavour.
So the rules are proved from the definition once and then used without further justification: the derivative of a constant is zero, differentiation respects sums and constant multiples, the product of f and g differentiates to f g′ + g f′, and the quotient f/g differentiates to (g f′ − f g′)/g² wherever g is not zero.
The power rule gives n x to the power n − 1.
A sign to watch in the notes
One set of lecture slides prints the quotient rule with a plus sign in the numerator, while another set gives it with the minus sign. The rule with the minus sign is the correct one, and the order of the two terms matters, because swapping them reverses the sign of the whole answer.
Starting the numerator with the denominator's function is a reliable way to remember the order.
The chain rule and why it is a product
If g is differentiable at x₀ and f is differentiable at g(x₀), the composite f of g has derivative f′(g(x₀)) times g′(x₀).
The lecture proof introduces two error terms, one for each function, shows that each tends to zero, and reads off the product of the two derivatives in the limit. In practice: differentiate the outside, keep the inside unchanged, and multiply by the derivative of the inside.
A second variable, as in y as a function of u with u a function of x, is the same rule in Leibniz form.
Slopes of curves that are not functions
An equation such as x² + y² = 4 defines y implicitly on suitable ranges. Differentiating every term with respect to x, with a factor y′ attached to every y term by the chain rule, and then solving for y′ gives the slope without solving for y.
Higher derivatives come by two routes in the notes: differentiate the expression for y′ and substitute it back, or differentiate the original equation repeatedly and eliminate the lower derivatives.
What this chapter covers
- 01
The constant, sum and multiple rules
- 02
The product rule and its proof
- 03
The quotient rule and its sign
- 04
The chain rule and the proof with two error terms
- 05
Intermediate variables in Leibniz form
- 06
Implicit differentiation of a curve
- 07
Higher derivatives by two methods
- 08
Derivatives of inverse functions
Worked example · free
A quotient with a root in the denominator
- 3Apply the quotient rule with the chain rule for the denominator.
- 2Combine the numerator over a single power of x² + 1.
- 1State the simplified derivative.
Key terms
- Product Rule
- The rule that the derivative of f times g is f times the derivative of g plus g times the derivative of f.
- Quotient Rule
- The rule that the derivative of f over g is g times the derivative of f minus f times the derivative of g, all over g squared, wherever g is not zero.
- Composite Function
- A function formed by applying one function to the output of another, differentiated with the chain rule.
- Second Derivative
- The derivative of the derivative, which measures how the rate of change is itself changing.
- Implicit Function
- A function defined by an equation linking x and y rather than by a formula for y in terms of x.
Rules of Differentiation and Implicit Differentiation FAQ
What is the difference between the chain rule and the product rule?
The product rule handles two functions multiplied together, each depending on x. The chain rule handles one function applied to the output of another. Many expressions need both: (x² + 1)(3x − 2) to the fourth power needs the product rule for the multiplication and the chain rule for the power of 3x − 2.
Why does implicit differentiation put y′ on every y term?
Because y is being treated as a function of x, so differentiating any expression in y with respect to x needs the chain rule. The derivative of y² with respect to x is 2y times y′, not 2y. Forgetting the factor y′ is the most common error in implicit differentiation and makes it impossible to solve for the slope.
How do I find the second derivative of an implicit curve?
Differentiate your expression for y′ with respect to x, using the quotient rule and attaching y′ wherever y appears, then substitute the expression for y′ back in. Finally look for the original equation in the numerator: it usually simplifies the result to a single term.
Exam move
Drill the three rules in mixed sets rather than one rule at a time, because the skill tested is choosing the right rule for each part of an expression. For every chain rule answer, write the derivative of the inside as a separate bracket before multiplying out, so that it cannot be lost.
Working through Rules of Differentiation and Implicit Differentiation in MTH6131? Sia is AskSia’s AI Mathematics tutor — ask any MTH6131 Rules of Differentiation and Implicit Differentiation question and get a clear, step-by-step explanation grounded in how MTH6131 is taught and assessed. Read this chapter free, then take your hardest questions to Sia.
