How to d/dx

y= (9+((x-9)/6)^1/5)^1/2 ?
Hope u can give me the way how to calculate it. ^^

I got 1/60 ((x-9)/6) ^ (-9/10), i think my answer is wrong because i check at wolfram website the ans is differrent from mine (sorry i didn't write the ans bcus the ans is quite long and difficult to type)

not quite ....

let u = ( (x-9)/6 )^(1/5)

then y = (9 + u)^1/2

dy/du = (1/2)(9+u)^(-1/2)

du/dx = (1/5)( (x-9)/6 )^(-4/5) (1/6)
= (1/30) ( (x-9)/6 )^(-4/5)

so dy/dx = dy/du * du/dx
= (1/2)(9 + u)^-1/2) (1/30) ( (x-9)/6 )^(-4/5)

= (1/60) (9 + ( (x-9)/6 )^(1/5)^(-1/2) ( (x-9)/6 )^(-4/5)

I suggest you check my steps for any errors, I should have written it out on paper first.
Wolfram seems to have performed some kind of complicated simplification.

How about this way:

square both sides:

y^2 = 9 + ( (x-9)/6 )^1/2 , then by implicit differentiation,

2y dy/dx = (1/2)( (x-9)/6) )^(-1/2)

dy/dx = (1/4)(√(6/(x-9) ) / y
or √6/(4y√(x-9) ) , that's not bad

not bad, but

y^2 = 9 + ((x-9)/6)^(1/5)
2y dy/dx = 1/5 ((x-9)/6)^(-4/5)
dy/dx = 1/(10y ((x-9)/6)^(4/5))

thanks for looking after me,

rotten copy errors.

thanks for the calculation but i think there is a mistake at implicit diff.

y^2= 9 + ((x-9)/6)^(1/5)
2y dy/dx = 1/5 ((x-9)/6)^(-4/5)) (1/6)
dy/dx = 1/60y ((x-9)/6)^(-4/5) ..... Sorryif im wrong

Btw why at wolfram the answer is
1/( 60y(-9+y^2)^4). Why is the x become y^2 and why there are no 6 and ^(4/5)?

To find the derivative of the given function:

Step 1: Identify the main function and any nested functions within it. In this case, the main function is y = (9 + ((x - 9)/6)^(1/5))^(1/2).

Step 2: Apply the chain rule to differentiate each nested function. The chain rule states that if we have a function g(f(x)), then the derivative is g'(f(x)) * f'(x), where g'(f(x)) is the derivative of the outer function and f'(x) is the derivative of the inner function.

Let's break down the given function and apply the chain rule step by step:

1. The outermost function is the square root, so we need to take the derivative of the square root function.

d/dx [√(u)] = (1/2) * (u^(-1/2)) * du/dx, where u is the expression within the square root.

2. The expression within the square root is (9 + ((x - 9)/6)^(1/5)), so we need to find its derivative.

Let's denote the expression (x - 9)/6 as a new variable, u = (x - 9)/6.

Using the power rule, the derivative of u^(1/5) is (1/5) * (u^(-4/5)) * du/dx.

3. Finally, we can substitute back into the derivative of the square root function using the chain rule:

d/dx [√(u)] = (1/2) * (u^(-1/2)) * du/dx = (1/2) * (9 + ((x - 9)/6)^(1/5))^(-1/2) * (1/5) * ((x - 9)/6)^(-4/5) * d/dx [((x - 9)/6)].

Now, we need to find d/dx [((x - 9)/6)].

Let's simplify the expression ((x - 9)/6) by multiplying the numerator and the denominator by 1/6:

((x - 9)/6) = (1/6) * (x - 9).

The derivative of (1/6) * (x - 9) with respect to x is simply (1/6).

4. Substituting this result back into the chain rule expression:

d/dx [((x - 9)/6)] = (1/6).

Putting it all together, the derivative of the given function is:

d/dx [y] = (1/2) * (9 + ((x - 9)/6)^(1/5))^(-1/2) * (1/5) * ((x - 9)/6)^(-4/5) * (1/6).

Simplifying further:

d/dx [y] = (1/2) * (1/5) * (1/6) * (9 + ((x - 9)/6)^(1/5))^(-1/2) * ((x - 9)/6)^(-4/5).

Combining the constants:

d/dx [y] = 1/60 * (9 + ((x - 9)/6)^(1/5))^(-1/2) * ((x - 9)/6)^(-4/5).

This is the derivative of the function y with respect to x. If you are typing the answer into a computer system, you can use the caret symbol (^) for exponentiation and parentheses to denote the different parts of the expression.