Calculus: Analytic Geometry and Calculus, with Vectors

3.6 The chain rule and differentiation of elementary functions 169

Problems 3.69

1 Calculate f'(xo) and write the equation of the line tangent to the graph of
y = f(x) at the point (xo,yo) when

(a) f(x)=x2, x0 =1 11ns.:y-1 =2(x1)

(b) f(x) = x(1 - x), xo = 0 Ans.: y = x

(c) AX) = ex, xo = 0 Ans.: y = x + 1

X

(d) f(x) =^1

+

x2, xo = 1 .4ns.: y = T

2 Become thoroughly familiar with the following technique, because it

enables us to do many chores quickly and correctly. Suppose we are required

to find dy/dx when

(1) y = sin 2x.

We must realize that we are not required to differentiate sin x but are required

to differentiate sin u, where u is a function of x. We look at (1) and read "y

equals sine u" and realize without making a lot of noise and without writing

anything that u = 2x. We then write dy/dx and say this is equal to cos u

(write cos 2x) times du/dx (write 2). When we follow orders, we get

(2)

dy

dx=(cos 2x)2,

but it is always better to put the answer in the neater form

(3) dydx= 2 cos 2x

which does not require parentheses.

3 Read the equations

(a) y = (x2 + 1)" (b) y = cos ex

(c) y = sin (ax + b) (d) y = e-

(e) y = cos ax (f) y = log (x2 + 1)

the way we read them when we want to find dy/dx. In the first case, we can

tolerate "y equals u to the nth" as a contraction of "y equals u with the exponent

n" or "y equals u to the nth power." In another case, we can tolerate "y equals

e to the u," which looks bad in print but is universally understood. Now, sup-

posing that n, a, and b are constants, concentrate upon the task of learning five

basic formulas and applying them to obtain the answers

(a) dx = 2nx(x2 + 1)n-1

(c) dx = a cos (ax + b)

= -a sin ax

(b) - = -ex sin ex

(d) dx = aeax

Ly 2x

(f)dx x2 + I

Practice the technique until the answers can be obtained quickly and effortlessly.