Find the derivative of the function.
q = square root of 18r-r^5
q = all this under square root
To find the derivative of the given function, q = √(18r - r^5), we can use the chain rule of differentiation.
Let's break it down step by step:
Step 1: Identify the function and its corresponding parts.
Our function is q = √(18r - r^5). Here, the outer function is the square root, and the inner function is 18r - r^5.
Step 2: Apply the chain rule.
The chain rule states that if we have a composite function y = f(g(x)), then its derivative dy/dx is given by dy/dx = f'(g(x)) * g'(x).
Step 3: Find the derivative of the outer function.
The derivative of the square root function is 1 / (2√x).
Step 4: Find the derivative of the inner function.
To find the derivative of 18r - r^5, we differentiate each term separately.
The derivative of 18r with respect to r is simply 18.
The derivative of -r^5 with respect to r can be found using the power rule.
The power rule states that if we have a term of the form f(x) = x^n, then its derivative f'(x) is given by f'(x) = n * x^(n-1).
So, for -r^5, the derivative with respect to r is -5r^(5-1) = -5r^4.
Step 5: Multiply the derivatives obtained in steps 3 and 4.
Multiply the derivative of the outer function (1 / (2√(18r - r^5))) with the derivative of the inner function (18 - 5r^4).
The final derivative of the function q = √(18r - r^5) is:
dq/dr = (1 / (2√(18r - r^5))) * (18 - 5r^4)
Simplifying further, we can rewrite the derivative as:
dq/dr = (18 - 5r^4) / (2√(18r - r^5))