Denition approach of learning new topics

28 4 DERIVATIVES
Example. A general non-mathematical example of composite function.
Rotation of fan (r) depends on the current passing into the fan (c), mathematically
can be written r = f(c)
And the ux of air (a) made to move by the fan on rotation (r), mathematically
can be written a = g(r)
This composite function current(c) → rotation(r) → air flux(a)
This same example can be converted to parametric form
Rotation of fan (r) depends on current(c) i.e. r = f(c)
Also air ux(a) coming from the fan depends on the current(c) i.e. a = g(c)
Example. Given f(x) = sin x & g(x) = x 2 then (f ◦ g)(x) & (g ◦ f)(x) ?
Solution :
(f ◦ g)(x) = f(g(x)) = f(x 2 ) = sin(x 2 )
(g ◦ f)(x) = g(f(x)) = g(sin x) = (sin x) 2 = sin 2 x
Example. Given f(x) & g(x) as in the above example nd (f ◦ g ◦ f ◦ g)(x)
(f ◦ g ◦ f ◦ g)(x) = f(g(f(g(x))))
= f(g(f(x 2 )))
= f(g(sin x 2 ))
= f(sin 2 x 2 )
= sin(sin 2 x 2 )
Problem. Practice problems
1. If f(x) = x 2 & g(x) = √ x then nd f ◦ g & g ◦ f
2. If f(x) = √ 1 + x 2 & g(x) = √ x 2 − 1 then nd f ◦ g & g ◦ f
4.8 Derivative of a composite function
If y = (f ◦ g)(x) is a composite of functions of f(x) & g(x) then
y = f(x), x = g(t) ⇒
y = f(g(t)) same as composite.
right!
(f ◦ g) ′ (x) = f ′ (g(x)) · g ′ (x)
d(f ◦ g)
dx
This is same as chain rule,
If y = f(x) & x = g(t) then
= df(g(x)
dx
dy
dt = dy
dx · dx
dt
· dg
dx
Example. If the displacement x(t) is given as a function of time t as x(t) =
2t 2 + 1
Instanteneous Velocity is v(t) = d dt (x(t)) = d dt (2t2 + 1) = 4t
Instanteneous Acceleration is a(t) = d dt (v(t)) = d dt (4t) = 4