The derivative of f(x) = cos(x) can be estimated using the approximation .The percentage error is calculated as . The percentage error in the derivative of f(x) at x = π/6 radian, choosing h = 0.1 radian, is

The correct option is > 0.1% and < 1%.

Step-by-Step Explanation:

1. Find the Exact Value of the Derivative:
Given the function:
$$f(x)=\cos (x)$$
Taking the derivative with respect to $x$:
$$f^{\prime }(x)=-\sin (x)$$
Now, we evaluate the exact derivative at $x=\frac{\pi }{6}$ radians:
$$f^{\prime }\left(\frac{\pi }{6}\right)=-\sin \left(\frac{\pi }{6}\right)=-0.5$$
Thus, the Exact Value is $-0.5$.

2. Find the Approximate Value using Central Difference Formula:
From the first image, the central difference approximation formula for the derivative is:
$$f^{\prime }(x)\approx \frac{f(x+h)-f(x-h)}{2h}$$
Substituting $f(x)=\cos (x)$ into the formula:
$$f^{\prime }(x)\approx \frac{\cos (x+h)-\cos (x-h)}{2h}$$
Using the trigonometric identity expansion:
$\cos (x+h)=\cos (x)\cos (h)-\sin (x)\sin (h)$
$\cos (x-h)=\cos (x)\cos (h)+\sin (x)\sin (h)$
Subtracting the two terms yields:
$$\cos (x+h)-\cos (x-h)=-2\sin (x)\sin (h)$$
Substituting this result back into the approximation formula:
$$f^{\prime }(x)\approx \frac{-2\sin (x)\sin (h)}{2h}=\frac{-\sin (x)\sin (h)}{h}$$
For the values $x=\frac{\pi }{6}$ radians and step size $h=0.1$ radian:
$$f^{\prime }\left(\frac{\pi }{6}\right)\approx \frac{-\sin \left(\frac{\pi }{6}\right)\sin (0.1)}{0.1}$$
Knowing that $\sin (0.1)\approx 0.0998334$:
$$f^{\prime }\left(\frac{\pi }{6}\right)\approx \frac{-0.5\times 0.0998334}{0.1}=-0.499167$$
Thus, the Approximate Value is approximately $-0.499167$ (or $-0.4992$ when rounded as in the third image).

3. Calculate the Percentage Error:
From the second image, the percentage error is given by:
$$\text{Percentage Error}=\left(\frac{\text{Exact value}-\text{Approximate value}}{\text{Exact value}}\right)\times 100$$
Substituting our calculated values:
$$\text{Percentage Error}=\left(\frac{-0.5-(-0.499167)}{-0.5}\right)\times 100$$
$$\text{Percentage Error}=\left(\frac{-0.000833}{-0.5}\right)\times 100\approx 0.167\%$$
Since $0.167\%$ is greater than $0.1\%$ and less than $1\%$, the percentage error falls within the range:
> 0.1% and < 1%