r(x) =( √x + 9+ √x)^5

Calculate the derivative of the function

To calculate the derivative of the function r(x) = (√x + 9 + √x)^5, we can use the chain rule. The chain rule states that if we have a function in the form h(g(x)), then its derivative is given by the product of the derivative of the outer function h'(g(x)) and the derivative of the inner function g'(x).

In this case, the outer function is h(x) = x^5, and the inner function is g(x) = √x + 9 + √x.

To find the derivative, we first need to find the derivative of the outer function h'(x), which is simply 5x^(5-1) = 5x^4.

Next, we need to find the derivative of the inner function g'(x). We can break down the inner function into three separate terms:

1. The derivative of √x with respect to x is (1/2)x^(-1/2).
2. The derivative of √x with respect to x is (1/2)x^(-1/2).
3. The derivative of 9 with respect to x is 0 since 9 is a constant.

Now we can calculate the derivative of the inner function by adding the derivatives of the three terms:

g'(x) = (1/2)x^(-1/2) + (1/2)x^(-1/2) + 0 = x^(-1/2).

Finally, we can multiply the derivative of the outer function with the derivative of the inner function to get the overall derivative:

r'(x) = h'(g(x)) * g'(x) = 5x^4 * x^(-1/2) = 5x^(4 - 1/2) = 5x^(7/2).

So, the derivative of the function r(x) = (√x + 9 + √x)^5 is r'(x) = 5x^(7/2).