Denition approach of learning new topics

20 4 DERIVATIVES
1. f(t) = sin t, g(t) = t + 1
t − 1
2. f(t − 1) = sin(t − 1), g(t) = t − 1 + 1
t − 1 − 1 = t
t − 2
3. f(e x ) = sin(e x ), g(e x ) = ex + 1
e x − 1
4. f(ax 2 + bx + c) = sin(ax 2 + bx + c), g(ax 2 + bx + c) = ax2 + bx + c + 1
ax 2 + bx + c − 1
5. f(cos x) = sin(cos x), g(x) = cos x + 1
cos x − 1
6. f(h(x)) = sin(h(x)), g(x) = h(x) + 1
h(x) − 1
Denition. Composite function
Given functions y = f(x) & y = g(x) then composition of functions f ◦g(x) =
f(g(x))
Example. Given f(x) = sin x & g(x) = e x then nd f ◦ g(x) & g ◦ f(x)
f ◦ g(x) = f(g(x)) = sin(g(x)) = sin(e x ) &
g ◦ f(x) = g(f(x)) = e f(x) = e sin x
Problem. Find the derivative of the function y = f(x) = x
dy
dx = lim f(x + h) − f(x)
h→0 h
x + h − x
= lim
h→0 h
= lim
h
= lim 1
h→0
= 1
h→0
h
That means the derivative of f(x) = x is d
dx (x) = 1
We know the derivative is slope of the tangent at a point x. Now derivative
1 means slope of the tangent to f(x) = x at any point is 1
Problem. Find the derivative of the function y = f(x) = x 2 .
Here in the problem they have not asked at which point of x we want to nd
the derivative means they want us to nd the derivative at any point x. Let us
use the above denition.