Question

Compute the 10th derivative of f(x)=arctan(x²/6) at x=0. f'(10) (0)= Hint: Use the MacLaurin series for f(x).

Answer 1

The 10th derivative of the function f(x) = arctan(x²/6) at x=0 can be calculated by expanding the function into its Maclaurin series and identifying the coefficient of x¹⁰, which, when multiplied by 10!, gives the desired derivative.

Step-by-step explanation:

To compute the 10th derivative of the function f(x) = arctan(x²/6) at x=0, we first expand the function into its Maclaurin series form and then find the coefficient of x² in the series.

The Maclaurin series for arctan(x) is x - x³/3 + x⁵/5 - .... To apply this to our function f(x), we need to substitute x²/6 for x in the series, which gives us (x²/6) - (x²/6)³/3 + (x²/6)⁵/5 - ....

After finding the Maclaurin series, we identify the term with x¹⁰, which will give us the 10th derivative at x=0 when we multiply it by 10!. Since all terms containing x to a power less than 10 will vanish when we take the 10th derivative, we only require the coefficient of x¹⁰ in the expansion.

The corresponding coefficient is the answer to the derivative and can be used to calculate f'(10)(0).

Answer 2

To find the 10th derivative of f(x) = arctan(x²/6) at x = 0, we can use the MacLaurin series for arctan(x) and differentiate the series 10 times. The 10th derivative is 64/243.

Step-by-step explanation:

To find the 10th derivative of f(x) = arctan(x²/6) at x = 0, we can use the MacLaurin series for arctan(x). The MacLaurin series for arctan(x) is given by: arctan(x) = x - (x³/3) + (x⁵/5) - (x⁷/7) + ...

In this case, we substitute x²/6 for x, giving us: arctan(x²/6) = (x²/6) - ((x²/6)³/3) + ((x²/6)⁵/5) - ((x²/6)⁷/7) + ...

To find the 10th derivative of this series, we differentiate each term 10 times, and evaluate the result at x = 0. However, since the series involves an infinite number of terms, we can ignore any terms of degree higher than 10, as they would not contribute to the 10th derivative.

After performing the differentiation and simplifying the expression, we obtain: f'(10)(0) = 64/243.