Question

Which among the following correctly defines Leibnitz rule of a function given by \( f (α) = \int_a^b (x,α)dx\) where a & b are constants?

(a) \(f’(α) = \frac{∂}{∂α}\int_a^b f (x,α) dx\)

(b) \(f’(α) = \frac{d}{dα} \int_a^b f (α) dx\)

(c) \(f’(α) = \int_a^b \frac{∂}{∂α} f (x,α) dx\)

(d) \(f’(α) = \int_a^b \frac{d}{dα} f (x,α) dx\)

I got this question during a job interview.

I'd like to ask this question from Differentiation Under Integral Sign topic in division Partial Differentiation of Engineering Mathematics

Answer 1

Correct option is (c) \(f’(α) = \int_a^b \frac{∂}{∂α} f (x,α) dx\)

For explanation: \(f’(α) = \int_a^b \frac{∂}{∂α} f (x,α) dx = \frac{d(f(α))}{dα} = \frac{d}{dα} \int_a^b f (x,α) dx.\)