Question

Let ( f ) be a function defined by the rule [ f(x)=3+2.9 tan ^-1(3.3 ln (x)) ] Compute the derivative and evaluate it at ( x=5 ). That is, compute ( f'(5) ).

Answer 1

Given the function f(x) = 3 + 2.9*atan(3.3*log(x)).

Step 1: Find the derivative of the function
We will use the chain rule to differentiate this function. The chain rule states that the derivative of a composite function is the derivative of the outer function times the derivative of the inner function. Here, the outer function is atan(u) and the inner function is 3.3*log(x).

The derivative of atan(u) with respect to u is 1/(u²+1). Thus, the derivative of atan(3.3*log(x)) with respect to x is 1/(3.3*log(x)² + 1) * 3.3/log(x). The multiplication by 2.9 gives 2.9*1/(3.3*log(x)² + 1)*3.3/log(x) = (9.57/(x*(10.89*log(x)² + 1))) .

The complete derivative f'(x) is then 0 + 9.57/(x*(10.89*log(x)² + 1)), the zero comes from the derivative of the constant term 3.

Step 2: Evaluate the derivative at x=5
To find the derivative at a particular x-value, we substitute that x-value into the derivative.

Hence, substituting x = 5 into the derivative gives f'(5) = 9.57/[5*(10.89*log(5)² + 1)] = 1.914/(1 + 10.89*log(5)²) .

So, the derivative of f at x = 5 is f'(5) = 1.914/(1 + 10.89*log(5)²).