Final answer:
To numerically estimate the derivative of tan(x²) at x = 0.1, one can use the finite difference method by choosing a small value for h and calculating the difference quotient.
Step-by-step explanation:
The numerical derivative of tan(x²) at x = 0.1 can be calculated using the definition of the derivative, which is the limit of the difference quotient as h approaches 0. However, because the question asks for a numerical derivative, we will use the concept of finite differences to approximate this value. The derivative f'(a) at some point a can be numerically estimated using the formula:
f'(a) ≈ ³−[f(a + h) - f(a)] / h
where h is a small number. Thus, for tan(x²) at x = 0.1, with a small value of h, say 0.0001, we perform the following computation:
- Calculate tan((0.1 + h)²).
- Calculate tan(0.1²).
- Subtract the value from step 2 from the value in step 1.
- Divide the result by h.
This will give us an approximation of the derivative of tan(x²) at x = 0.1.
Remember that this is an approximation and the exact value would require calculus techniques involving limits.