Solução_Calculo_Stewart_6e

F.
130 ¤ CHAPTER 3 DIFFERENTIATION RULES
TX.10
(f ) The Quotient Rule: If f and g are both differentiable, then d
dx
f(x)
g(x) d
d
f(x) − f(x)
= dx dx g(x)
g(x)
[ g(x)] 2 .
The derivative of a quotient of functions is the denominator times the derivative of the numerator minus the numerator
times the derivative of the denominator, all divided by the square of the denominator.
(g) The Chain Rule: If f and g are both differentiable and F = f ◦ g is the composite function defined by F (x) =f(g(x)),
then F is differentiable and F 0 is given by the product F 0 (x) =f 0 (g(x)) g 0 (x). The derivative of a composite function is
the derivative of the outer function evaluated at the inner function times the derivative of the inner function.
2. (a) y = x n ⇒ y 0 = nx n−1 (b) y = e x ⇒ y 0 = e x
(c) y = a x ⇒ y 0 = a x ln a (d) y =lnx ⇒ y 0 =1/x
(e) y =log a x ⇒ y 0 =1/(x ln a) (f ) y =sinx ⇒ y 0 =cosx
(g) y =cosx ⇒ y 0 = − sin x (h) y =tanx ⇒ y 0 =sec 2 x
(i) y =cscx ⇒ y 0 = − csc x cot x (j) y =secx ⇒ y 0 =secx tan x
(k) y =cotx ⇒ y 0 = − csc 2 x (l) y =sin −1 x ⇒ y 0 =1/ √ 1 − x 2
(m) y =cos −1 x ⇒ y 0 = −1/ √ 1 − x 2 (n) y =tan −1 x ⇒ y 0 =1/(1 + x 2 )
(o) y =sinhx ⇒ y 0 =coshx (p) y =coshx ⇒ y 0 =sinhx
(q) y =tanhx ⇒ y 0 =sech 2 x (r) y =sinh −1 x ⇒ y 0 =1/ √ 1+x 2
(s) y =cosh −1 x ⇒ y 0 =1/ √ x 2 − 1 (t) y =tanh −1 x ⇒ y 0 =1/(1 − x 2 )
e h − 1
3. (a) e is the number such that lim =1.
h→0 h
(b) e =lim
x→0
(1 + x) 1/x
(c) The differentiation formula for y = a x [y 0 = a x ln a] is simplest when a = e because ln e =1.
(d) The differentiation formula for y =log a x [y 0 =1/(x ln a)] issimplestwhena = e because ln e =1.
4. (a) Implicit differentiation consists of differentiating both sides of an equation involving x and y with respect to x,andthen
solving the resulting equation for y 0 .
(b) Logarithmic differentiation consists of taking natural logarithms of both sides of an equation y = f(x), simplifying,
differentiating implicitly with respect to x, and then solving the resulting equation for y 0 .
5. (a) The linearization L of f at x = a is L(x) =f(a)+f 0 (a)(x − a).
(b) If y = f(x), then the differential dy is given by dy = f 0 (x) dx.
(c)SeeFigure5inSection3.10.