Question

The third derivative of f(x) = cos(x)sinh(x) ⁄ x at x = 0 is

(a) 0

(b) ^π⁄32

(c) (π)^2

(d) cos(1)sinh(1)

I had been asked this question by my school teacher while I was bunking the class.

Question is from Leibniz Rule topic in chapter Differential Calculus of Engineering Mathematics

Answer 1

Right option is (a) 0

The explanation is: Assume y = f(x)

Rewriting the part sinh(x)⁄x as infinite series we have

\(\frac{sinh(x)}{x}=\frac{1}{1!}+\frac{x^2}{3!}+\frac{x^4}{5!}….\infty\)

Now the function f(x) becomes

y=\(cos(x)(\frac{1}{1!}+\frac{x^2}{3!}+\frac{x^4}{5!}….\infty)\)

Taking the third derivative of the above function using Leibniz rule we have

y^(3)=\(c_0^3sin(x)(\frac{1}{1!}+\frac{x^2}{3!}+\frac{x^4}{5!}….\infty)-c_1^3cos(x)(\frac{2x}{3!}+\frac{4x^3}{5!}….\infty)\) \(-c_2^3sin(x)(\frac{2}{3!}+\frac{12x^2}{5!}….\infty)+c_3^3cos(x)(\frac{24x}{5!}…\infty)\)

Now substituting x = 0 we have

y^(3) = 0.