Question

Use Maclaurin series to evaluate each of the following. Although you could do them by computer, you can probably do them in your head faster than you can type them into the computer. So use these to practice quick and skillful use of basic series to make simple calculations.

$\frac{{\mathbf{d}}^{\mathbf{10}}}{{\mathbf{dx}}^{\mathbf{10}}}\left(x^8{\tan }^{2}x\right)\mathbf{}\mathbf{at}\mathbf{}\mathbf{x}\mathbf{=}\mathbf{0}\mathbf{.}$

Short answer

The value of the derivative is 3628800 .

Step-by-step solution

Given Information

The derivative is $\frac{d^{10}}{d{x}^{10}}\left(x^8{\tan }^{2}x\right)$.

Definition of the Maclaurin Series

A Maclaurin series is the series that gives the sum of the function's derivatives. The formula used here is mentioned below as:

${\mathbf{tan}}^{\mathbf{2}}\left(x\right)\mathbf{=}{\mathbf{x}}^{\mathbf{2}}\mathbf{+}\frac{\mathbf{2}{\mathbf{x}}^{\mathbf{4}}}{\mathbf{3}}\mathbf{+}\frac{\mathbf{17}{\mathbf{x}}^{\mathbf{6}}}{\mathbf{45}}\mathbf{+}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}$

Find the value

The formula is mentioned below.

${\tan }^2\left(x\right)=x^2+\frac{2x^4}{3}+\frac{17x^6}{45}+....$

The function becomes as follows.

$x^8{\tan }^2\left(x\right)=x^{10}+\frac{2x^{12}}{3}+\frac{17x^{14}}{45}+....$

Derivate the above equation ten times, and substitute the value of x. The derivative becomes as follows.

$$\begin{gathered} \frac{d^{10}}{d{x}^{10}}\left(x^8{\tan }^2\left(x\right)\right)=10\times 9\times 8\times 9\times 6\times 5\times 4\times 3\times 2\times 1 \\ =3628800 \end{gathered}$$

The value of the derivative is$3628800$ .