Find d.c of the following functions with respect to x.
1. y= sin^2(3x+4)
2.y= x.cosx/1+x^2
Answers
Answer:
Differentiate f(x) = (x3+1)2.
The only way we have of doing this so far is by first multiplying out the brackets and then differentiating. If we do this we get
f(x) = x6 + 2x3 +1 and therefore f´(x) = 6x5 +6x2.
This is no problem with a simple example such as the one above but what happens if we have for example f(x) = (x3+1)6 ?
In this case it takes far too much effort to multiply out the brackets before differentiating.
To differentiate composite functions like this we use what is called the Chain Rule. We'll do example 1 again to see how it works.
f(x) is an example of a composite function as was introduced in functions 2.
It can be written as f(u) = u2 where u = x3+1, u is a function of x, that is u(x) = x3+1.
The Chain rule says that we first differentiate f(u) regarding u as the variable and get f´(u) = 2u ( just as (x2)´ = 2x )
Next we differentiate u and get u´(x) = 3x2. Finally we multiply the two results together and get
f´(x) = 2u·3x2. Putting back the value of u we get f´(x) =2 ( x3+1)·3x2 = 6x5 +6x2
This gives us a rule called the Chain Rule which says that