`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/
3 Answers
hkj1385 | (Level 1) Assistant Educator
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)]
tiffanynelson | In Training Educator
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))`
balajia | College Teacher | (Level 1) eNoter
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))`