Taking logarithms first, when the base and the power both vary

What to assume they know

• Theorem 5: the derivative of the natural exponential and of the natural logarithm
• The convention, fixed on Part I p. 127, that an unsubscripted logarithm is the natural one
• The logarithm of a product, a quotient and a power, from §5.4
• The chain rule and the product rule
• Implicit differentiation, and the permission to differentiate an unknown function
• Where a trigonometric expression is positive, on a stated interval
• Collecting an unknown derivative from several terms and solving for it

What they should be able to do

• Recognise the two shapes that call for logarithms before differentiating
• Take logarithms of a relation and differentiate the result, producing the derivative divided by the function on the left
• Derive the general formula for a variable power raised to a variable exponent and read it term by term
• State the positivity requirement and identify the inputs a given item needs it to exclude
• Differentiate a long product or quotient by turning it into a sum of logarithms
• Differentiate a constant raised to the variable power, by two routes
• Split a sum of several variable powers before taking logarithms, and say why the splitting is necessary
• Handle a relation in which the unknown function appears in an exponent, and collect the derivative
• Supply the missing domain for an exercise item whose base can be negative

Where it usually goes wrong

• "Logarithmic differentiation is a new differentiation rule." It is a rewriting followed by the rules already in hand. Nothing is being assumed that §5.4 did not supply.
• "The positivity condition is a technicality." It is the condition under which the second line of the derivation exists at all. Exercise 5.5 Q3 and Q7 are undefined unless the input exceeds one, and the printed items say nothing.
• "Take logarithms and then differentiate the left side to get the derivative." The left side gives the derivative divided by the function. Forgetting to multiply back is the single commonest error in this section.
• "You can take logarithms of a sum." You cannot usefully. Example 30 splits the sum into named summands first and takes logarithms of each. Q12 and Q11 of the Miscellaneous Exercise both turn on this.
• "A constant to the variable power needs logarithms." It does not — Example 28's second route rewrites the constant as a natural exponential and applies the chain rule. Logarithms are the shorter path, not the only one.
• "The answer should be free of the original expression." For a variable power it normally is not. Example 29 prints the answer with the original function as a factor, and then expands it into two terms which each still carry a power of the input.
• "If the base can be negative the method just fails." The method needs a restriction, not an abandonment. Supply the interval on which the base is positive and proceed; the Miscellaneous Exercise items show the chapter doing exactly that.
• "Q17's three methods might give different answers." They cannot, and the question is asking the student to confirm it. The value of the item is that logarithms turn a page of product rule into four lines.

Questions to check understanding

• Differentiate a variable base raised to a variable exponent — the form of Exercise 5.5 Q3, Q4, Q7, Q8, Q9 and Q11
• Differentiate a long product or quotient by taking logarithms — the form of Exercise 5.5 Q1, Q2 and Q5
• Differentiate a constant raised to the variable power by two different routes — the form of Example 28
• Differentiate a sum of two or more variable powers, splitting first — the form of Exercise 5.5 Q12 and Miscellaneous Exercise Q11
• Differentiate a relation in which variable powers appear on both sides and collect the derivative — the form of Exercise 5.5 Q13 and Q14
• Evaluate the derivative of a four-factor product at one input — the form of Exercise 5.5 Q16
• State the interval on which a given item's logarithm exists, and justify it

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 shape §5.5 opens on (Part I p. 130). A function written as one expression raised to the power of another, both depending on the input. Say out loud why the existing rules fail: the power rule wants a constant exponent, and the exponential derivative of Theorem 5 wants a constant base. This item has neither, and it is the only shape in the chapter for which no earlier rule applies at all.
• The derivation of the general formula (Part I p. 130). Take logarithms of both sides; the exponent comes down as a factor. Differentiate: the left gives the derivative divided by the function, by the chain rule with the logarithm outside; the right needs the product rule and gives two terms. Multiply through by the function. Verified: the two right-hand terms are the exponent times the base's derivative over the base, and the exponent's derivative times the logarithm of the base. Both terms should be named — one is the contribution of the moving base and the other of the moving exponent, and every later item is these two terms with different contents.
• The positivity sentence (Part I p. 130). The chapter states that the whole expression and the base must both be positive, or their logarithms have no meaning. Verified as necessary: the derivation takes the logarithm of both sides at the second line, which requires the left side positive, and produces the logarithm of the base on the right, which requires the base positive. A printed slip in the sentence immediately after this one is recorded in Notes.
• Example 27 (Part I p. 131). A square root of a quotient with three quadratic or linear factors. Taking logarithms halves everything and turns the quotient into a difference. Verified: the logarithm of the whole is one half of the sum of the logarithms of the two numerator factors minus the logarithm of the denominator; differentiating gives one half of the reciprocal of the first factor, plus twice the input over the second, minus the derivative of the third over the third; and multiplying back by the original expression finishes it. This is the item to run in full — it is the clearest demonstration that the technique is about representation, not about powers.
• Example 28 (Part I p. 131). A positive constant raised to the variable power. The chapter does it twice: once by taking logarithms, and once by rewriting the constant as the natural exponential of its own logarithm and using the chain rule. Verified: both give the function itself times the logarithm of the constant. The second route is worth the time because it shows that logarithmic differentiation is a convenience here rather than a necessity, and because the same rewriting is the standard way to handle any constant base.
• Example 29 (Part I p. 132). The input raised to the sine of the input, for positive inputs. Verified: the general formula gives the function times the bracket containing the sine over the input plus the cosine times the logarithm, and expanding the bracket gives two terms — the input to the power one less than the sine, times the sine; plus the original function times the cosine times the logarithm. The chapter prints both the bracketed and the expanded forms, and the expanded one is what shows the two contributions separately.
• Example 30 (Part I pp. 132–133). Three variable powers summed and set equal to a constant. The chapter names the three summands, differentiates the sum to zero, then handles each summand separately by logarithms, and finally collects. Verified: the first summand's logarithm brings the input down as a factor multiplying the logarithm of the unknown function, so its derivative carries the unknown derivative inside; the second is the mirror image; the third is the input to its own power and produces the familiar one-plus-logarithm factor. The collection at the end is a linear solve. Four numbered intermediate results are carried on the page and referred back to; keep the numbering in view, because the final line is unreadable without it.
• Exercise 5.5 Q1 to Q11 (Part I p. 134). Eleven differentiations. Verified, in outline: Q1 is a product of three cosines and gives the product times minus the sum of the tangent, twice the tangent of twice the input, and three times the tangent of three times the input. Q2 is a square root of a five-factor quotient and gives half the expression times the alternating sum of five reciprocals. Q3, Q4, Q7, Q8, Q9 and Q11 each contain at least one variable power and reduce to the general formula. Q5 is a triple product of powers and gives the expression times two over the first factor plus three over the second plus four over the third. Q6 and Q10 mix a variable power with an ordinary quotient, so the two halves are differentiated by different methods and added. Q5 is the best first exercise — no variable power at all, and it is four lines by logarithms against a page by the product rule.
• Exercise 5.5 Q12 to Q15 (Part I p. 134). Four relations to differentiate. Verified: Q12 sums two variable powers to a constant and is Example 30 with one summand removed. Q13 sets one variable power equal to the other and gives, after taking logarithms of both sides and collecting, the second letter times the quantity itself minus the first letter times the logarithm of the second, all over the first letter times the first letter minus the second times the logarithm of the first. Q14 has cosines as both bases and yields a quotient of two brackets, each a logarithm plus a tangent term. Q15 is anomalous and is recorded in Notes.
• Exercise 5.5 Q16, Q17 and Q18 (Part I p. 134). Q16 is a product of four bracketed factors whose derivative is wanted at one specific input. Q17 asks for the same derivative by three named methods and whether they agree. Q18 asks for the three-factor product rule to be established twice, once by repeated application and once by logarithms. Verified for Q16: the logarithmic route gives the sum of four terms, which at the input one become one half, one, two and four; their sum is fifteen halves; the product itself is sixteen there; and the derivative is therefore one hundred and twenty. That single number is worth showing, because the direct expansion is a degree-fifteen polynomial. Q17 and Q18 are method-comparison items and belong at the end of the explanation, not in the drill.
• Miscellaneous Example 41 (Part I p. 143). The sine raised to the sine, on the interval where the sine is positive. Verified: the general formula gives the function times the cosine times one plus the logarithm of the sine. Note that the chapter attaches the interval here — which is exactly what Exercise 5.5 never does.
• Miscellaneous Exercise Q7, Q9, Q10 and Q11 (Part I pp. 144–145). Q7 is a logarithm raised to a logarithm, above the input one. Q9 is a difference of sine and cosine raised to itself, on a stated half-turn. Q10 is a sum of four terms mixing a variable base, a variable exponent and two constants. Q11 is a sum of two variable powers, above the input three. Verified: Q7 gives the function times one plus the logarithm of the logarithm, all over the input. Q9 gives the function times the sum of sine and cosine times one plus the logarithm of the base. Q10 gives the input to its own power times one plus the logarithm, plus the constant times the input to one less than the constant, plus the constant to the input times the logarithm of the constant — and the fourth term differentiates to zero, which is the point of including it. Q11 needs the general formula twice and added. All four carry intervals, and section 5 should say so.

Figures to have open

• A three-panel comparison for section 1: constant exponent, constant base, and both varying. Not in the book; §5.5 prints no figure at all, and neither does any page from Part I p. 129 to Part I p. 146 — verified on the page image of every page.
• An annotated version of the general formula for section 4, with the two right-hand terms colour-separated and labelled. The formula is the chapter's; the annotation is added here.
• A table for section 5 listing the Exercise 5.5 items whose bases can be negative or zero, against the interval each needs. Use the repo's DataTable component. This table does not exist in the book — the exercise attaches no interval to any item — and building it is the most useful thing the explanation can do for a student.
• A worked build-up of Example 27 for section 7, with the three logarithm terms appearing one at a time above the line and their derivatives appearing beneath.
• No redraw of any textbook figure is possible in this topic, because there is none in the section.

Where this sits in the book

• NCERT Class 12 Mathematics, Chapter 5, §5.5 Logarithmic Differentiation, the general derivation and the positivity requirement, Part I p. 130
• Examples 27 and 28, Part I p. 131; Example 29, Part I p. 132; Example 30, Part I pp. 132–133
• Exercise 5.5, questions 1 to 18, Part I p. 134
• Miscellaneous Example 41, Part I p. 143; Miscellaneous Exercise on Chapter 5, questions 7, 9 and 10, Part I p. 144, and question 11, Part I p. 145
• Summary, the logarithmic differentiation bullet, Part I p. 146