Question

For the given function f(x)=\(\sqrt{x^3+x^7}\) the values of first and second derivative at x = 1 are assumed as 0 and 1 respectively. Then the value of the third derivative could be

(a) 54√2

(b) 2√2

(c) √2

(d) Indeterminate

I had been asked this question in a job interview.

The origin of the question is Leibniz Rule topic in division Differential Calculus of Engineering Mathematics

Answer 1

The correct answer is (a) 54√2

The explanation: Rewriting the given function as

y^2 = x^3 + x^7

Now applying the Leibniz rule up to the third derivative we have

(y^2)^(3)=(x^3+x^7)^(3)

\(c_0^3y^{(3)}y+c_1^3y^{(2)}y^{(1)}+c_2^3y^{(1)}y^{(2)}+c_3^3yy^{(3)}=3!+(7.6.5)x^4\)….(1)

Equating both sides and substituting x = 1 we get

y^(1) = 0

Now assumed in the question are the values y^(1) = 0 and y(2) = 1

We also know y(1) = √2

Putting them in equation (1) we get

2√2 y^(3) = 3! + 210 = 216

y^(3) = 54√2.