Calculus: Analytic Geometry and Calculus, with Vectors

3.6 The chain rule and differentiation of elementary functions^171

10 Supposing that a and w (omega, to keep physicists and engineers happy)
are constantsand

show how the formula

Q = eat sin wt,

dQ

dt = eal(w cos wt) + (sin wt)aead

= ea,(w cos wt + a sin wt)

is obtained. Then let I = dQ/dt and show that

dl

dt = ea'[2aw cos wt + (a2 - w2) sin wt].

Remark: It is not necessary for us to know that, if a < 0, Q might be the charge
on the capacitor of an LRC oscillator, in which case the electric current would be
I and the voltage drop across the inductor would be the product of dI/dt and the
inductance L of the inductor. It is, however, a good idea to know that the things
we are learning are important in applied mathematics.
11 Prove that
d ex - ez 2 _\2.
dxex+e s- ez+e z

12 If, for a positive integer n,

yn(x) = sin x+sin 2x+sin 3x+ +sin nx

1 2 3 n '

show that

y;,(x) = cosx+ cos 2x+ cos 3x + + cosnx.

13 Calculate f'(x) from the first and then from the second of the formulas

AX) = logI

1 +

-x I , f ( x )

Make the results agree. Hint: Do not forget the second formula in (3.676);

the derivative with respect to x of log Jul is (1/u) du/dx and the absolute-value

signs quietly disappear.

14 Observe that if y is a differentiable function of x, so also is the function

F having values

F(x) = x2 + xy(x) + [y(x)]2.

Tell precisely what formulas are used to obtain the formula

F'(x) = 2x + xy'(x) + y(x) + 2y(x)y'(x)

dx=2x+xdx+y+2ydx

fins.: The power formula, the formula for the derivative of a product, the chain

formula, and the formula for the derivative of a sum.