Question

Use the chain rule to find the derivative of

f(x)=√(x²+81/x²+64)
Use sqrt() for √(x). Simplify you answer.
fʹ(x)=__

Answer 1

The derivative of the function f(x) = sqrt((x² + 81) / (x² + 64)) is found by expressing the square root as a power of 1/2, applying the chain rule to find the derivative of the inner and outer functions, and multiplying them together. The quotient or product rule is used for the derivative of the inner function, and the final expression for f'(x) should be simplified.

Step-by-step explanation:

To find the derivative of the function f(x) = sqrt((x² + 81) / (x² + 64)) using the chain rule, we first express the square root as a power of 1/2: f(x) = ((x² + 81) / (x² + 64))^(1/2). Next, we apply the chain rule, which involves taking the derivative of the outer function and then multiplying it by the derivative of the inner function. The function inside the square root (or to the power of 1/2) is u(x) = (x² + 81) / (x² + 64), so we need to find u'(x). Let's go through the steps:

1. Derivative of the outer function: If g(x) = x^(1/2), then g'(x) = (1/2)x^(-1/2). So, f'(x) with respect to u is (1/2)u^(-1/2).
2. Derivative of the inner function u(x): First, we write the inner function as u(x) = (x² + 81)(x² + 64)^(-1), then apply the quotient rule or product rule accordingly to find u'(x).
3. Multiply the derivative of the outer function by the derivative of the inner function to get f'(x).

Finally, we simplify the expression to get the simplified form of f'(x).