4.6 Theorems On Derivatives [ Differentiation Rules ]
1) If u and v are differentiable functions of x, then
In general the derivative of the sum ( or difference ) of two or more functions w.r. to x is the sum ( or difference ) of their derivatives w.r. to x.
2) If u and v are differentiable functions of x, then
Corollary: If k is
constant and f (x) is a differentiable function of x, then
Corollary:
3) If u and v are differentiable functions of x, then
Corollary :
Proofs : 1) Let y = u ±
v
Let D
y, D u
and D v
be changes in y, u and v respectively corresponding to a change
D x in x.
Then y + Dy
= ( u + Du)
± ( v +
Dv )
subtracting, we get, Dy
= Du ±
Dv
2) Let y = u.v
Let Dy,
Du, Dv
be changes in y, u and v respectively corresponding to a small change
Dx in x.
