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Chapter 5 · Continuity and Differentiability

Exercise 5.7 · Q2: Second derivative of x20x^{20}

Exercise 5.7Second order derivative · Section 5.7
The question

Find the second order derivatives of the functions given in Exercises 1 to 10.

2. x20x^{20}

Exam answer

Let y=x20y = x^{20}⁠. Then

dydx=20x19\frac{dy}{dx} = 20x^{19}

∴ d2ydx2=20⋅19x18=380x18\therefore\ \dfrac{d^2y}{dx^2} = 20\cdot 19x^{18} = 380x^{18}

Want the reason behind every step, the basics and the checks?

The one idea this problem is built on

The power rule ddxxn=nxn−1\frac{d}{dx}x^n = nx^{n-1} brings the power down as a factor and lowers the power by one. Using it twice brings down nn and then n−1n - 1⁠:

d2dx2xn=n(n−1) xn−2.\frac{d^2}{dx^2}x^n = n(n-1)\,x^{n-2}.

With n=20n = 20 that is 20⋅19 x18=380x1820 \cdot 19\,x^{18} = 380x^{18}⁠.

Given, to find, and the plan
Given
y=x20y = x^{20}⁠, defined and differentiable for every real xx⁠.
To find
d2ydx2\dfrac{d^2y}{dx^2}⁠.
Plan
(1) State the domain. (2) Power rule once for dydx\frac{dy}{dx}⁠. (3) Constant-multiple rule and power rule again for d2ydx2\frac{d^2y}{dx^2}⁠.

Basics you need

1Power rule

ddxxn=nxn−1\frac{d}{dx}x^n = nx^{n-1} (Table 5.3). For a whole number nn it holds for every real xx⁠. Basics: derivative rules

2Constant multiple rule

ddx(c u)=c dudx\frac{d}{dx}(c\,u) = c\,\frac{du}{dx} for a constant cc⁠: the constant waits in front while uu is differentiated.

3Second order derivative (Section 5.7)

d2ydx2=ddx(dydx)\frac{d^2y}{dx^2} = \frac{d}{dx}\left(\frac{dy}{dx}\right)⁠. Lesson: second order derivative

Solution

  1. Name the function and its domain

    Let y=x20y = x^{20}⁠. A whole-number power of xx is a polynomial, defined and differentiable for every real xx⁠, so the answer holds for all x∈Rx \in \mathbf{R}⁠.

    There is no denominator, root or logarithm, so no value of xx has to be excluded.

  2. First derivative: power rule with n=20n = 20

    dydx=ddx(x20)=20x20−1=20x19.\frac{dy}{dx} = \frac{d}{dx}\left(x^{20}\right) = 20x^{20-1} = 20x^{19}.
    The power 20 comes down in front as a factor, and the new power is one less, 19.
    Power rule (Table 5.3). Basics: derivative rules

  3. Second derivative: constant multiple rule, then power rule with n=19n = 19

    d2ydx2=ddx(20x19)=20 ddx(x19)=20⋅19x18=380x18.\begin{aligned} \frac{d^2y}{dx^2} &= \frac{d}{dx}\left(20x^{19}\right) = 20\,\frac{d}{dx}\left(x^{19}\right) \\ &= 20 \cdot 19x^{18} \\ &= 380x^{18}. \end{aligned}

    So d2ydx2=380x18\dfrac{d^2y}{dx^2} = 380x^{18} for every real xx⁠.

    We differentiate the first derivative 20x1920x^{19} from Step 2. The constant 20 stays in front (constant multiple rule) and only x19x^{19} is differentiated: its power 19 comes down and the power drops to 18. Finally 20×19=38020 \times 19 = 380⁠.
    Section 5.7: d2ydx2=ddx(dydx)\frac{d^2y}{dx^2} = \frac{d}{dx}\left(\frac{dy}{dx}\right)⁠. Lesson: second order derivative

  4. A quick look at the sign

    x18=(x9)2≥0x^{18} = (x^9)^2 \ge 0⁠, so d2ydx2=380x18≥0\frac{d^2y}{dx^2} = 380x^{18} \ge 0 for every xx⁠, and it is zero only at x=0x = 0⁠. The slope 20x1920x^{19} therefore never decreases as xx increases: the graph of x20x^{20} bends upward, like a very flat-bottomed parabola.

    This is not needed for the answer, but reading the sign of a second derivative is the main use of it in Chapter 6 (Application of Derivatives).
Result

For y=x20y = x^{20} (all real xx⁠):

dydx=20x19,d2ydx2=380x18.\frac{dy}{dx} = 20x^{19}, \qquad \frac{d^2y}{dx^2} = 380x^{18}.

(Textbook answer: 380x18380x^{18}⁠.)

Check it with numbers

At x=1x = 1⁠: d2ydx2=380⋅118=380\frac{d^2y}{dx^2} = 380 \cdot 1^{18} = 380⁠, and 20⋅19=38020 \cdot 19 = 380 is exactly the "bring down 20, then 19" count.

At a=1.1a = 1.1⁠: the formula gives 380(1.1)18≈2112.769380(1.1)^{18} \approx 2112.769⁠. The second difference with h=0.001h = 0.001⁠,

y(1.101)−2y(1.1)+y(1.099)0.000001≈2112.813,\begin{aligned} &\frac{y(1.101) - 2y(1.1) + y(1.099)}{0.000001} \\ &\approx 2112.813, \end{aligned}

agrees to four significant figures (the small gap is the error of the approximation, which shrinks like h2h^2⁠). Also y(1.101)−y(1.099)0.002\frac{y(1.101) - y(1.099)}{0.002} is about 122.324122.324⁠, against 20(1.1)19≈122.31820(1.1)^{19} \approx 122.318⁠. ✓

Common mistakes
  1. Forgetting to lower the power the second time: 380x19380x^{19} is wrong; each differentiation lowers the power by one, so two of them give x18x^{18}⁠.
  2. Multiplying by 20 twice: 400x18400x^{18} is wrong. The second factor is the new power, 19.
  3. Confusing d2ydx2\frac{d^2y}{dx^2} with (dydx)2=400x38\left(\frac{dy}{dx}\right)^2 = 400x^{38}⁠.
What to take away
  1. d2dx2xn=n(n−1)xn−2\frac{d^2}{dx^2}x^n = n(n-1)x^{n-2}⁠: the power and the next lower number both come down.
  2. Constants stay in front while you differentiate.

Every fact used, and where it comes from

FactUsed inSource
ddxxn=nxn−1\frac{d}{dx}x^n = nx^{n-1}Steps 2, 3Table 5.3, Section 5.3 (Basics: derivative rules)
ddx(cu)=cdudx\frac{d}{dx}(cu) = c\frac{du}{dx}Step 3Algebra of derivatives (Class XI), Section 5.3
d2ydx2=ddx(dydx)\frac{d^2y}{dx^2} = \frac{d}{dx}\left(\frac{dy}{dx}\right)Step 3Section 5.7 (Lesson: second order derivative)
Even powers are ≥0\ge 0Step 4x18=(x9)2x^{18} = (x^9)^2
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