Question

y = root(3)(1 + 4x)      Write the composite function in the form f(g(x)). Identify the inner function u = g(x) and the outer function y =f(u).  Then find the derivative dy/

Accepted Answer

Note:- 1) if y = x^n ; then dy/dx = n*x^(n-1)
Thus,
If y = (1+4x)^(1/3) ; then 
Let f(x) = x^(1/3)...........(1)
And g(x) = (1+4x).........(2)
Thus, f(g(x)) = (1+4x)^(1/3)  .................Answer
Now, dy/dx = y' = (1/3)*{(1 + 4x)^(-2/3)}*4
or, dy/dx = (4/3)*[(1+4x)^(-2/3)]

Answer

Let it be noted that y is equivalent to (1+4x)^(1/3)
We see that the outer function y=f(u)= u^(1/3)
We see that the inner function u+g(x)=1+4x
The derivative of the inner function g'(x)=4
The derivative of the outer function f'(u)= 
This is equivalent to (1/3)u^(-2/3)
To find the derivative of y, we simply multiply the two derivatives that we just found, f'(u) and g'(x) together and we replace the u with x. 
This gives 4(1/3)x^(-2/3)
This can also be written as  (4)/(3x^(2/3))

Answer

Let y=f(g(x))
y'=f'(g(x)).g'(x)
In the given question y=(1+4x)^(1/3).
Here f(x)=(x)^(1/3) and g(x)=1+4x
y'=(1/3)(1+4x)^(1/3-1).(4)
=4/(3(1+4x)^(2/3))

Math

Write The Composite Function In The Form F(g(x)). [identify The Inner Function U = G(x) And The Outer Function Y = F(u).]

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