MATH1851 Chap.6 Differentiation Rules and Implicit Curves
Differentiation Rules and Implicit Curves
Structure chooses the rule
Before differentiating, identify the outermost operation. A sum uses termwise rules, a product needs two changing-factor terms, a quotient needs two complete products over a squared denominator, and a composite needs the outer derivative evaluated at the unchanged inside times the inner derivative.
Reading structure first prevents a familiar symbol from triggering the wrong rule.
Product and quotient rules preserve both factors
The product rule arises because changing a rectangle's two sides creates two first-order strips. Differentiating both factors and multiplying them misses that geometry. The quotient rule carries the same two contributions with a subtraction and a denominator square.
Labelling numerator and denominator before substitution keeps the minus sign attached to complete products.
The chain rule multiplies nested rates
A composite changes in two stages. The input changes the inner value, and that inner change moves through the outer function. Work from outside inward while preserving each inside until its turn.
Count the nontrivial layers before writing symbols; the count predicts how many derivative factors should appear.
Implicit differentiation follows every y term
When x and y are tied by a relation, differentiate both sides with respect to x. Every derivative taken through y gains a dy over dx factor by the chain rule. Product terms such as xy also require the product rule.
Collect all derivative terms, factor dy over dx, solve and substitute a point only after the symbolic formula is complete.
Audit the finished derivative
A product derivative should show both factors taking a turn, a quotient should retain a squared denominator, and a composite should contain an inner derivative factor. After implicit work, every differentiated y term must carry dy over dx.
These structural checks catch omissions before algebraic simplification hides them.
What this chapter covers
- 01
Constant, power, sum and trigonometric rules
- 02
Product-rule structure and verification
- 03
Quotient-rule signs and domain restrictions
- 04
Nested layers in the chain rule
- 05
Implicit differentiation workflow
- 06
Higher derivatives and repeating trig cycles
Worked example · free
Differentiate an implicit curve and find its tangent
- 4Differentiate every term, using product and chain rules where required.
- 3Collect and solve all derivative terms symbolically.
- 2Check the point, evaluate the slope and write the tangent line.
Key terms
- Product Rule
- The derivative rule producing one term for each factor changing while the other is retained.
- Quotient Rule
- The derivative rule combining numerator and denominator changes over the square of the denominator.
- Chain Rule
- The derivative of an outer function at the inner output multiplied by the inner derivative.
- Implicit Relation
- An equation connecting x and y without first isolating y as one explicit function.
- Higher Derivative
- A derivative taken repeatedly, such as the second derivative describing change in the first derivative.
- Domain Restriction
- An excluded input inherited from the original function and retained after differentiation or simplification.
Differentiation Rules and Implicit Curves FAQ
How do I know which differentiation rule to use first?
Look at the outermost operation. A product of two large expressions uses the product rule even if each factor contains a composite. A power or trigonometric function applied to another expression uses the chain rule. Work from the outside inward.
Why does the product rule have two terms?
Both factors change with the input. One first-order contribution changes the first factor while retaining the second, and the other changes the second while retaining the first. Multiplying the two derivatives ignores those contributions and does not match the limit derivation.
When do I attach dy over dx in implicit differentiation?
Attach it whenever differentiating a function of y with respect to x. For y squared, the result is two y times dy over dx. In a product such as x y, use the product rule, producing x times dy over dx plus y.
Exam move
Annotate each expression with its outer structure before calculating. Use temporary labels for complicated numerators, denominators and inner functions. After differentiating, run a structure audit: products should contribute two terms, quotients should retain a squared denominator, composites should carry every inner factor and implicit y terms should carry dy over dx.
Verify selected products by expansion and selected composites by a numerical slope check. For higher trigonometric derivatives, track the four-step function cycle separately from coefficient growth.
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