Big Derivatives.

Calculus Level 2

If $f(x)=\sin(x^3)$ then what is $f^{(15)}(0)$ equal to?

Notation: $f^{(n)}(x)$ denotes the $n$ th derivative of $f(x)$ .

\frac{10!}{5!}

\frac{15!}{5!}

\frac{14!}{5!}

\frac{20!}{5!}

\frac{25!}{5!}

1 solution

Chew-Seong Cheong
Sep 26, 2018

$\begin{aligned} f(x) & = \color{#3D99F6} \sin (x^3) & \small \color{#3D99F6} \text{By Maclaurin series} \\ & = \color{#3D99F6} x^3 - \frac {x^9}{3!} + \frac {x^{15}}{5!} - \cdots \\ \implies f^{(15)} (x) & = \frac {15!}{5!} - \frac {18! x^3}{3! 6!} + \frac {21! x^6}{6! 7!} - \cdots \\ f^{(15)} (0) & = \boxed{\dfrac {15!}{5!}} \end{aligned}$

@Hana Wehbi , it is better to use $f^{(15)} (0)$ and mention that "where $f^{(n)}(x)$ denotes the $n$ th derivative of $f(x)$ ". Because $f^3(x)$ can also mean $(f(x))^3$ or $f(f(f(x)))$ . Strictly, "If ...." is not full sentence. It should not end with a period but a comma.

Chew-Seong Cheong - 2 years, 8 months ago

Ok, l will. Thank you for pointing this.

Hana Wehbi - 2 years, 8 months ago

Big Derivatives.

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