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

To find the percentage error in the estimated derivative, we will calculate both the exact value and the approximate value of the derivative of $f\left(x\right)=cos\left(x\right)$ at $x=\frac{\pi }{6}$ with $h=0.1$ radian.

1. Exact Value of the Derivative:
The exact derivative of $f\left(x\right)=cos\left(x\right)$ is:
$f^{\prime }\left(x\right)=-sin\left(x\right)$
Evaluating this at $x=\frac{\pi }{6}$:
${\text{Exact Value}}_{}=-sin\left(\frac{\pi }{6}\right)=-0.5$

2. Approximate Value of the Derivative:
Using the central difference formula shown in the first image:
$f^{\prime }\left(x\right)\approx \frac{f(x+h)-f(x-h)}{2h}$
Substitute $f\left(x\right)=cos\left(x\right)$:
$f^{\prime }\left(x\right)\approx \frac{cos(x+h)-cos(x-h)}{2h}$
Using the trigonometric identity $cos\left(A\right)-cos\left(B\right)=-2sin\left(\frac{A+B }{2}\right)sin\left(\frac{A-B}{2}\right)$, we simplify the numerator:
$cos(x+h)-cos(x-h)=-2sin\left(x\right)sin\left(h\right)$
Now substitute this back into the approximation formula:
${\text{Approximate Value}}_{}=\frac{-2sin\left(x\right)sin\left(h\right)}{2h}=-sin\left(x\right)\frac{sin\left(h\right)}{h}$
Substitute the values $x=\frac{\pi }{6}$ and $h=0.1$ radian:
${\text{Approximate Value}}_{}=-sin\left(\frac{\pi }{6}\right)\frac{sin\left(0.1\right)}{0.1}=-0.5\times \frac{sin\left(0.1\right)}{0.1}$
Using the value of $sin\left(0.1\right)\approx 0.0998334166$:
${\text{Approximate Value}}_{}\approx -0.5\times \frac{0.0998334166}{0.1}=-0.5\times 0.998334166=-0.499167083$

3. Percentage Error:
As given in the second image, the percentage error is:
$\text{Percentage Error}=\left(\frac{\text{Exact value}-\text{Approximate value}}{\text{Exact value}}\right)\times 100$
Substitute the calculated values into the formula:
$\text{Percentage Error}=\left(\frac{-0.5-(-0.499167083)}{-0.5}\right)\times 100$
$\text{Percentage Error}=\left(\frac{-0.000832917}{-0.5}\right)\times 100\approx 0.00166583\times 100=0.1666\%$

Since the percentage error is approximately 0.167%, it satisfies the condition:
> 0.1% and < 1%