Differentiating twice, and what the second derivative is for

What to assume they know

• Every differentiation rule from earlier in the chapter, including the chain, product and quotient rules
• The derivatives in Table 5.3 and the two in Theorem 5
• The three inverse trigonometric derivatives from §5.3.3
• Implicit differentiation, and the permission to differentiate an unknown function
• The parametric derivative formula and its proviso, from §5.6
• Rearranging an equation to clear a square root from a denominator
• The modulus function written in two branches

What they should be able to do

• Differentiate a derivative and write the result in each of the five notations the chapter gives
• Say what has to be true of the first derivative before a second one exists
• Compute the second derivative of a product, a quotient and a composite
• Verify that a stated function satisfies a stated relation among it and its first two derivatives
• Clear a square root by multiplying through before differentiating a second time
• Express a second derivative in terms of the function's own output rather than its input
• Compute a second derivative for a curve given parametrically, dividing twice
• Compute a second derivative for a function defined in two branches, and say what happens at the join

Where it usually goes wrong

• "The second derivative of a parametric pair is the second parameter derivative divided by the first." It is not. Differentiate the first ratio with respect to the parameter, then divide again by the first coordinate's parameter derivative. Miscellaneous Exercise Q17 fails immediately under the wrong rule.
• "A function with a first derivative has a second." Only if the first is itself differentiable, which the section states in a conditional clause. The cube of a modulus has two derivatives and no third at the join.
• "The exercise is asking me to compute something." Six of its seventeen items ask you to confirm a stated relation. That is a different task with a different write-up, and the section never introduces it.
• "The second derivative tells you about maxima and minima." True, and not in this chapter — the words do not appear here at all. Do not import the test.
• "The five notations mean five different things." They are five spellings of one object. Students meeting the subscripted form for the first time in Example 38 often take it for something new.
• "You can differentiate a quotient with a square root twice head-on." You can, and it is unpleasant. Both routes through Example 38 clear the radical first, and every hard item in the block does the same.
• "An answer must be in terms of the input." Exercise 5.7 Q12 asks explicitly for the output instead, and the substitution is legitimate only because of the branch the inverse cosine lives on.
• "Verification items have no method." They have a very fixed one: differentiate as far as the relation needs, substitute, and collect. Show the column layout once and every item becomes mechanical.

Questions to check understanding

• Compute the second derivative of a stated function — the form of Exercise 5.7 Q1 to Q10
• Verify that a stated function satisfies a stated relation among it and its first two derivatives — the form of Exercise 5.7 Q11 and Q13 to Q17
• Express a second derivative in terms of the output rather than the input — the form of Exercise 5.7 Q12
• Compute the second derivative of a curve given parametrically — the form of Miscellaneous Exercise Q17
• Show that a stated function has a second derivative at every input, and find it — the form of Miscellaneous Exercise Q18
• Show that a stated combination of the first two derivatives of a circle is constant — the form of Miscellaneous Exercise Q15
• Write the same second derivative in each of the five notations the chapter gives

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 whole of §5.7 (Part I p. 137). Ten printed lines. Set the output equal to a function of the input; the first derivative is numbered as an equation; provided that first derivative is itself differentiable, differentiate it again; the result gets a name and five notations — the doubled differential form, the double-primed function form, the operator form, the double-primed output form and the subscripted form. The section closes by saying higher orders follow the same pattern. The conditional clause is the only mathematics on the page and section 2 is built on it.
• Example 35 (Part I p. 138). A cube plus a tangent. Verified: the first derivative is three times the square plus the square of the secant; the second is six times the input plus twice the square of the secant times the tangent, and the middle step needs the chain rule on the squared secant. The chapter prints the intermediate form before collapsing it, which is the step students skip.
• Example 36 (Part I p. 138). Two constants times a sine and a cosine, shown to satisfy a relation. Verified: the second derivative is the negative of the original function, so the sum is zero. Note the change of task — nothing is being computed for its own sake; a stated identity is being confirmed. This is the first of the chapter's verification items and section 4 exists to name the change.
• Example 37 (Part I p. 138). A combination of two exponentials with different multipliers, shown to satisfy a relation among the function and its first two derivatives. Verified: the first derivative is six times the sum of the two exponentials; the second is twelve times the first plus eighteen times the second; and substituting into the stated combination cancels every term. Write the cancellation out as a column — three rows of coefficients summing to zero in each exponential — because that layout is what makes the six verification items later in the exercise routine.
• Example 38 (Part I p. 139). The inverse sine, shown to satisfy a relation. The chapter gives two routes. Verified, first route: the first derivative is the reciprocal of the square root of one minus the square; multiply through by that square root to clear it; differentiate the product; the derivative of the square root supplies a factor that turns into the input over the square root; and multiplying through by the square root once more gives the stated relation. Verified, second route: square the cleared relation first, so that no square root survives, then differentiate the product and divide by twice the first derivative. The second route is shorter and it is where the subscript notation first appears; run both.
• Exercise 5.7 Q1 to Q10 (Part I p. 139). Ten straightforward second derivatives. Verified, in order: Q1 gives two. Q2 gives three hundred and eighty times the eighteenth power. Q3 gives minus twice the sine minus the input times the cosine. Q4 gives minus the reciprocal of the square. Q5 gives the input times six times the logarithm plus five. Q6 gives the exponential times ten times the cosine of five times the input minus twenty-four times the sine of it. Q7 gives the exponential times twenty-seven times the cosine of three times the input minus thirty-six times the sine of it. Q8 gives minus twice the input over the square of one plus the square. Q9 gives minus one plus the logarithm, over the square of the input times its logarithm. Q10 gives minus the sum of the sine and cosine of the logarithm, over the square of the input. Q3, Q5, Q6 and Q7 need the product rule at the second step, and Q6 and Q7 need it twice.
• Exercise 5.7 Q11 to Q17 (Part I pp. 139–140). Six verification items and one that is not. Verified: Q11 is Example 36 with numbers. Q12 asks for the second derivative of the inverse cosine in terms of the output alone, and the answer is the negative of the cotangent of the output times the square of its cosecant — obtained by writing the input as the cosine of the output and the square root of one minus its square as the sine of the output, which is legitimate because the inverse cosine's outputs lie where the sine is not negative. Q13 is three times the cosine plus four times the sine, both of a logarithm, and the stated relation holds; the coefficients cancel in pairs. Q14 is a sum of two exponentials with different rates, and the stated relation holds for every choice of the two multipliers. Q15 is the same shape with equal and opposite rates, and the second derivative is forty-nine times the function. Q16 rearranges to the negative logarithm of one more than the input, whose second derivative is the square of its first. Q17 is the square of an inverse tangent, and the relation follows by clearing the denominator once and differentiating again. Q12 is the odd one out and deserves its own section; every other item asks for a relation, and it alone asks for a change of variable in the answer.
• Miscellaneous Exercise Q18 (Part I p. 145). The cube of a modulus, whose second derivative is to be shown to exist everywhere and then found. Verified: on the non-negative side the function is the cube and on the negative side it is the negated cube, so the first derivative is three times the square on one side and minus three times the square on the other, and both give zero at the join; the second derivative is six times the input on one side and minus six times it on the other, and both give zero at the join. The answer is six times the modulus, which is a tidy way to say it and the chapter does not. A further fact, which is added here and should be flagged as such: that answer is itself a modulus scaled, so it fails to be differentiable at the join — the function has a second derivative everywhere and no third derivative at zero.
• Miscellaneous Exercise Q15 (Part I p. 145). A circle written by its equation, with a stated combination of the first and second derivatives to be shown constant and independent of the two centre coordinates. Verified: differentiate implicitly once to get the first derivative in terms of the two offsets; differentiate again to express the second offset through the second derivative; substitute both back into the circle's own equation; and the offsets disappear, leaving the radius. This is the chapter's only geometric use of a second derivative and it does not say so — the expression is the reciprocal of the curvature, and naming it is the explanation's addition.
• Miscellaneous Exercise Q17 and Q22 (Part I p. 145). Q17 is a parametric pair whose second derivative is wanted. Q22 is an exponential of a constant times an inverse cosine, to be shown to satisfy a relation among it and its first two derivatives. Verified: in Q17 the two parameter derivatives reduce to the constant times the parameter times the cosine and the same with the sine, so the first derivative is the tangent, and dividing its parameter derivative by the first parameter derivative again gives the cube of the secant over the constant times the parameter. Dividing twice, not differentiating twice, is the whole trick, and getting it wrong is the commonest error in the item. In Q22 the route is Example 38's second one exactly: clear the square root, square, and differentiate again.

Figures to have open

• A stacked notation card for section 1 showing all five printed spellings. The content is the chapter's; the layout is added here. §5.7 prints no figure — verified on the page image of every page from Part I p. 129 to Part I p. 146, none of which carries one.
• A coefficient table for section 5 built from Example 37. Use the repo's DataTable component.
• A two-branch graph of the cubed modulus for section 12, with its first and second derivatives drawn beneath it on shared axes, so that the second derivative's own corner at the join is visible. Not in the book, and it is the one drawing in this topic that earns its place.
• A sorting table of Exercise 5.7's first ten items for section 9, with a column naming the rule each second step needs. Use the repo's DataTable component.
• No redraw of any textbook figure is possible in this topic, because §5.7 has none.

Where this sits in the book

• NCERT Class 12 Mathematics, Chapter 5, §5.7 Second Order Derivative, the whole section, Part I p. 137
• Examples 35, 36 and 37, Part I p. 138; Example 38 with both routes, Part I p. 139
• Exercise 5.7, questions 1 to 11, Part I p. 139, and questions 12 to 17, Part I p. 140
• Miscellaneous Exercise on Chapter 5, questions 15, 17, 18 and 22, Part I p. 145
• Summary, Part I p. 146, which does not mention this section — see Notes