Question

Given that f(x)=x¹2 h(x), h(-1)=5, and hʹ(-1)=8, calculate fʹ(-1).
fʹ(-1)=__

Answer 1

To calculate f'(-1), we use the product rule of differentiation on f(x)=x² h(x). At x=-1, we deal with complex numbers. The result is f'(-1) = (5/2i) + 8i.

Step-by-step explanation:

To calculate f'(x) for f(x)=x¹² h(x) at x = -1, we need to use the product rule for differentiation, which states that if f(x)=u(x)v(x) then f'(x)=u'(x)v(x)+u(x)v'(x). Given h(-1)=5 and h’(-1)=8, the derivative of f(x) at x = -1 is:

f'(x) = (x¹²)'·h(x) + x¹²·h’(x)

Calculating the first term:

(x¹²)' = (√(x))' = 1/(2√(x))

Now evaluating at x = -1:

(-1¹²)' = 1/(2√(-1)) = 1/(2i) (since √(-1) = i)

For the second term at x = -1:

-1¹²·h’(-1) = i·(8) = 8i

Therefore,

f'(-1) = (1/(2i))·5 + 8i = (5/2i) + 8i

This problem involves complex numbers, as the exponent in the original function indicates taking the square root, which for negative numbers involves 'i', the imaginary unit. Therefore, the calculation of the derivative includes complex arithmetic.