Curves given through a parameter, and the derivative recovered by the chain rule

What to assume they know

• The chain rule, and the composite it is applied to here
• Derivatives of the sine, the cosine, the tangent and a general whole power
• The half-angle identities for the sine and cosine of a doubled angle
• The identity relating the squares of the sine and cosine
• Rearranging an equation of three factors to isolate one of them
• Reading an interval attached to a parameter, and what it excludes
• The derivative of an exponential with a constant base, from §5.5

What they should be able to do

• Say what a parametric form is and identify the parameter in a stated pair
• Derive the parametric derivative formula from the chain rule in one rearrangement
• State the proviso and say which parameter values it excludes for a given pair
• Differentiate a stated parametric pair and simplify the ratio
• Use a half-angle identity to reduce a ratio of two trigonometric expressions
• Explain why an answer left in the parameter is complete, quoting the chapter's own remark
• Choose a parametrisation for a curve given by an equation in the two letters, and verify that it satisfies the equation
• Convert a parametric answer back into the two original letters where the substitution allows it
• Differentiate one function with respect to another by taking a ratio of two derivatives

Where it usually goes wrong

• "You have to eliminate the parameter first." The exercise instruction says the opposite in so many words, and the boxed Note says an answer in the parameter is finished. Eliminating is usually harder and sometimes impossible.
• "The formula is a new rule." It is the chain rule divided through by one of its factors. Nothing is assumed beyond §5.3.1.
• "The proviso never matters." It excludes two parameter values in Miscellaneous Example 42, and the chapter stops the working to say so. Check it whenever the first coordinate's derivative can vanish.
• "Any answer containing the parameter is incomplete." The chapter's own boxed Note is the refutation, and every answer in Exercise 5.6 is of that form.
• "A parametrisation is unique." Example 34 chooses one and checks it. Any pair satisfying the equation would serve, and the derivative would come out the same at each point of the curve.
• "Cancelling a common factor is always safe." Not where the factor can vanish. Example 33's half-angle cancellation and Example 42's bracket cancellation both need a restriction, and only the second one is stated.
• "Differentiating one function with respect to another is a different technique." Miscellaneous Example 43 is the parametric formula with the input as the parameter.
• "A parametric pair always traces a curve." Exercise 5.6's second item, as printed, has the same trigonometric function in both coordinates, so the pair traces a straight segment and the derivative is a constant. See Notes.

Questions to check understanding

• Differentiate a stated parametric pair and simplify the ratio — the form of Exercise 5.6 Q1 to Q6 and Q9 to Q10
• Reduce a parametric ratio using a half-angle or triple-angle identity — the form of Exercise 5.6 Q3, Q6 and Q7
• State the parameter values at which a stated parametric derivative fails, and justify them from the proviso
• Propose a parametrisation for a curve given by an equation in the two letters and verify that it satisfies the equation — the form of Example 34
• Prove a stated closed form for a parametric derivative — the form of Exercise 5.6 Q11
• Differentiate one named function with respect to another — the form of Miscellaneous Example 43
• Carry a parametric first derivative on to a second derivative — the form of Miscellaneous Exercise Q17

Examples worth working on the board

Values marked verified are worked out here from the chapter's own printed data; neither answers file was opened, and this chapter prints no answers to its exercises.

• The opening of §5.6 (Part I pp. 134–135). It contrasts the parametric case with the two the chapter has already handled — one coordinate given directly by the other, and the two tangled together — and then names the third quantity. Verified as a three-way contrast: §5.3.2 supplied the other two on Part I p. 122, so the explanation can put all three side by side with one line each.
• The derivation of the formula (Part I p. 135). Apply the chain rule to get the derivative with respect to the parameter as a product of two derivatives, then divide. A logical point worth thirty seconds is recorded in Notes: the chain rule step already treats the second coordinate as a differentiable function of the first, which is exactly what is being computed.
• The proviso (Part I p. 135). The formula is stated twice, once with the bracketed condition that the first coordinate's derivative with respect to the parameter be non-zero, and once in a form naming the two derivatives with the same condition attached in square brackets. The chapter attaches it both times, which is unusually careful for this chapter.
• Example 31 (Part I p. 135). Both coordinates given by a constant times a cosine and a sine of the same angle. Verified: the two parameter derivatives are the constant times the negated sine and the constant times the cosine, and the ratio simplifies to the negated cotangent. Say what curve this is — the chapter does not — because naming it makes the answer's meaning visible at once.
• Example 32 (Part I p. 135). The first coordinate a constant times the square of the parameter, the second twice the constant times the parameter. Verified: the derivatives are twice the constant times the parameter and twice the constant, and the ratio is the reciprocal of the parameter. The constant cancels entirely, which is worth pointing at: the answer does not depend on which member of the family the curve is.
• Example 33 (Part I p. 136). A pair built from the parameter plus its sine, and one minus its cosine. Verified: the ratio is the sine over one plus the cosine, which the double-angle identities turn into the tangent of half the parameter — write the numerator as twice a half-angle sine times a half-angle cosine and the denominator as twice the square of the half-angle cosine, then cancel. The cancellation needs the half-angle cosine to be non-zero and the chapter does not say.
• The boxed Note (Part I p. 136). It observes that the answer is expressed through the parameter alone, without either of the two original letters. This is the permission slip for the whole exercise set and it should be quoted early rather than at its printed position, because a student who meets Exercise 5.6 without it will try to eliminate the parameter in every item.
• Example 34 (Part I p. 136). A curve given by an equation in the two coordinates with two-thirds powers. The chapter proposes a parametrisation with cubes of a cosine and a sine, checks that it satisfies the equation, and then differentiates. Verified: raising the cubes to the two-thirds power gives squares, which sum to one, so the proposed pair does lie on the curve; the two parameter derivatives carry a common factor of three times the constant times a sine times a cosine; and the ratio is the negated tangent, which the chapter then rewrites as the negated cube root of the second coordinate over the first. A printed bracket fault in the verification line is recorded in Notes.
• Exercise 5.6 Q1 to Q11 (Part I p. 137). Eleven parametric pairs. Verified, in order: Q1 gives the square of the parameter. Q2 is anomalous and is recorded in Notes; as printed it gives the ratio of the two constants, a pure number. Q3 gives minus four times the sine of the parameter, after the double-angle identity. Q4 gives minus the reciprocal of the square of the parameter. Q5 gives the cosine minus twice the doubled-angle cosine, over twice the doubled-angle sine minus the sine. Q6 gives the negated cotangent of half the parameter. Q7 is the hardest item in the set and reduces, after the triple-angle identities, to the negated cotangent of three times the parameter. Q8 gives the tangent of the parameter, after the logarithm-of-a-tangent term is differentiated and the two pieces combine into a single squared cosine over a sine. Q9 gives the ratio of the two constants times the cosecant. Q10 gives the tangent of the parameter, because both parameter derivatives carry a common factor of the constant times the parameter. Q11 asks for a stated answer and gives the negated ratio of the two coordinates, which follows because taking logarithms turns each coordinate into a constant multiple of an inverse trigonometric function and the two inverse derivatives are exact negatives of each other. Q7 and Q8 deserve their own worked passes; the other nine are drill.
• Miscellaneous Example 42 (Part I pp. 143–144). Both coordinates built from the parameter plus its reciprocal, one as an exponent on a constant base and one as a base under a constant exponent. Verified: both parameter derivatives carry the same factor of one minus the reciprocal of the square, and that factor cancels; what survives is the first coordinate's exponential times the logarithm of the constant, over the constant times the bracket to one less than the exponent. The chapter stops to say the cancellation is only legitimate away from two parameter values, which is section 10's whole content and the only place in the chapter where the proviso is exercised rather than quoted.
• Miscellaneous Example 43 (Part I p. 144). Differentiate one function with respect to another: the square of a sine against an exponential of a cosine. Verified: both are differentiated with respect to the input and divided; the sine cancels between numerator and denominator, and the answer is minus twice the cosine over the exponential. This is the same formula with the input playing the parameter's part, which is worth saying, because it turns a separate-looking technique into the section's own.
• Miscellaneous Exercise Q12 (Part I p. 145). A cycloid-shaped pair with different constants on the two coordinates and an interval on the parameter. Verified: the ratio is six fifths times the cotangent of half the parameter, by the same half-angle move as Example 33.
• Miscellaneous Exercise Q17 (Part I p. 145). A pair built from the cosine plus the parameter times the sine, and the sine minus the parameter times the cosine, with a second derivative wanted. Verified: the two parameter derivatives simplify beautifully to the constant times the parameter times the cosine and the constant times the parameter times the sine, so the first derivative is the tangent; differentiating the tangent with respect to the parameter and dividing again by the first parameter derivative gives the cube of the secant over the constant times the parameter. The second-derivative half belongs to the next topic, which carries it too; here it is the closing demonstration that a parametric first derivative feeds straight into a second.

Figures to have open

• A parameter-to-coordinates diagram for section 1: one node feeding two, with the induced relation between the two coordinates drawn as a dashed link. Use the repo's Network component. The chapter prints no figure in §5.6 — verified on the page image of every page from Part I p. 129 to Part I p. 146, none of which carries one.
• A traced circle for section 4 with the parameter shown as a turning angle and the two coordinate projections marked. Not in the book; naming the curve is the point.
• A parameter line for sections 3 and 10 with the forbidden values struck out, used twice so the second use reads as the first one instantiated.
• An eleven-row table of Exercise 5.6 for section 9, with columns for the two parameter derivatives and the simplified ratio. Use the repo's DataTable component. The items are the chapter's; the table is added here.
• No redraw of any textbook figure is possible in this topic, because §5.6 has none.

Where this sits in the book

• NCERT Class 12 Mathematics, Chapter 5, §5.6 Derivatives of Functions in Parametric Forms, the opening contrast, Part I pp. 134–135
• The derivation of the formula with its proviso, and Examples 31 and 32, Part I p. 135
• Example 33, the boxed Note and Example 34, Part I p. 136
• Exercise 5.6, questions 1 to 11, Part I p. 137
• Miscellaneous Examples 42 and 43, Part I pp. 143–144; Miscellaneous Exercise on Chapter 5, questions 12 and 17, Part I p. 145