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8th Edition

Stewart + 3 others

Publisher: Cengage Learning

ISBN: 9781305279148

Chapter 3.2, Problem 1PT

Textbook Problem

(*f*(*x*)*g*(*x*))′ = *f′*(*x*)*g*′(*x*).

Expert Solution

To determine

Whether the statement “

**Formula used:**

*General Power Rule:*

For any real numbers *n*, the derivative of the function

*Product Rule:*

If *f* and *g* are differentiable, then

**Calculation:**

Check whether the statement

From the above product rule, it is clear that

That is,

*Counter Example:*

Consider,

Compute

Study Guide for Stewart's Single Variable Calculus: Early Transcendentals, 8th

Show all chapter solutions

Ch. 3.1 - For f(x) = 6x4, f′(x) = _____.
24x
6x3
10x3
24x3
Ch. 3.1 - For y=12gt2 (where g is a constant), dydt=. a)...Ch. 3.1 - For f(x) = 5 + g(x), f(x) = _____. a) 5g(x) b) 5 +...Ch. 3.1 - limh0ah1h=1 for a = a) 0 b) 1 c) e d) The limit is...Ch. 3.1 - For f(x) = ex x, f(x) = a) ex x b) ex 1 c) e1 x...Ch. 3.1 - A particle moves along an axis so that after t...Ch. 3.2 - (f(x)g(x)) = f(x)g(x).Ch. 3.2 - For f(x) = (x3 + 1) (x4 + 1), f(x) = a) (3x2)(4x3)...Ch. 3.2 - For f(x)=xx+1,f(x)=. a) 1 b) x2(x+1)2 c) 1(x+1)2...Ch. 3.2 - For f(x)=exx,f(x)= a) ex b) xex1x2 c) exxexx2 d)...

Ch. 3.3 - limx0sin5x3x=. a) 1 b) 53 c) 35 d) 0Ch. 3.3 - If limx0sinxx=1, the angle x must be measured in:...Ch. 3.3 - For y = x sin x, y = _____. a) x cos x b) x cos x...Ch. 3.3 - For y = 3x + tan x, y = _____. a) 3 + sec x b) 3x...Ch. 3.3 - For y = sin2 x + cos2 x, y = _____. a) 2 sin x 2...Ch. 3.3 - What is the equation of the tangent line to y =...Ch. 3.4 - If y = f (g(x)) then y′ = f′(g′(x)).
Ch. 3.4 - For y = (x3 + 1)2, y = _____. a) 2x(x3 + 1) b)...Ch. 3.4 - For y = sin 6x, dydx=. a) cos 6x b) 6 cos x c) cos...Ch. 3.4 - For y=43+2x, y = _____. a) 4(3 + 2x)3/2 b) 4(3 +...Ch. 3.4 - For y = f (u) and u = g(x), dydx=dydududx.Ch. 3.4 - Sometimes, Always, or Never: (ax) = ax.Ch. 3.4 - If f(x) = 2e3x, f(x) = a) 2e3x b) 2xe3x c) 6e3x d)...Ch. 3.5 - y=4x2+1 defines y implicitly as a function of x.Ch. 3.5 - If y is defined as a function of x by the equation...Ch. 3.5 - The slope of the line tangent to the circle x2 +...Ch. 3.5 - The range of f(x) = tan1 x is: a) [2,2] b) (2,2)...Ch. 3.5 - sec1(23)= a) 3 b) 3 c) 6 d) 6Ch. 3.5 - For f(x) = cos1(2x), f(x) = a) 114x2 b) 214x2 c)...Ch. 3.6 - For y = ln(3x2 + 1), y= a) 13x2+1 b) 6x c) 63x2+1...Ch. 3.6 - For y = x3 ln x, y = a) x2(1 + 3 ln x) b) x3 + 3x2...Ch. 3.6 - For y = 3x, y = a) 3x log3 e b) 3x ln 3 c) 3xlog3e...Ch. 3.6 - By logarithmic differentiation, if y=xx+1, then y...Ch. 3.6 - limx0+(1+x)1/x= a) e b) e c) 1e d) 1eCh. 3.7 - True or False: f(t) is used to measure the average...Ch. 3.7 - Air is being pumped into a chamber so that after t...Ch. 3.7 - A bacteria population grows in such a way that...Ch. 3.7 - A circle is increasing in area at 4 cm2/min. How...Ch. 3.8 - The general solution to dydt=ky is: a) y(t) =...Ch. 3.8 - A bacteria culture starts with 50 organisms and...Ch. 3.8 - If $5,000 is deposited in an account that...Ch. 3.9 - A right triangle has one leg with constant length...Ch. 3.9 - How fast is the angle between the hands of an...Ch. 3.10 - For f(x) = x3 + 7x, df = _____. a) (x3 + 7x) dx b)...Ch. 3.10 - The best approximation of y using differentials...Ch. 3.10 - The linearization of f(x)=1x at x = 4 is: a)...Ch. 3.10 - Using differentials, an approximation to (3.04)3...Ch. 3.11 - sinx=ex+ex2Ch. 3.11 - cosh 0 = a) 0 b) 1 c) e22 d) is not definedCh. 3.11 - For f (x) = sinh x, f (x) = a) cosh2 x b) sinh2 x...Ch. 3.11 - For f(x)=x2+sinhx, f(x) = a) 2x+coshx2x2+sinhx b)...Ch. 3.11 - For f(x) = tanh1 2x, f(x) = a) 2(sech1 2x)2 b)...

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