Evaluating the 10th Derivative of ( sqrt{1-x^2} ) Using Taylor Series Expansion

In this article, we will explore how to evaluate the 10th derivative of the function ( sqrt{1-x^2} ) at ( x0 ) using Taylor series and binomial coefficients. We will start by discussing the Taylor series expansion of ( sqrt{1-u} ) and then apply it to find the required derivative.

The Taylor Series for ( sqrt{1-u} )

Consider the function ( f(u) sqrt{1-u} ). This function can be expanded using the binomial series:

1u^{1/2} sum_{k0}^{infty} binom{1/2}{k} u^k

Applying this to ( f(x) sqrt{1-x^2} ), we have:

sqrt{1-x^2} sum_{k0}^{infty} binom{1/2}{k} x^{2k}

The coefficient of ( x^{10} ) in this series is ( binom{1/2}{5} ), which corresponds to the 5th term in the series when ( k5 ).

Taylor Series and Binomial Coefficients

The Taylor series expansion of ( sqrt{1-u} ) at zero is given by:

sqrt{1-u} sum_{k ge 0} binom{1/2}{k} u^k

For ( ux^2 ), we get:

sqrt{1-x^2} sum_{k ge 0} binom{1/2}{k} x^{2k}

The Taylor series is also given by:

sqrt{1-x^2} sum_{n ge 0} frac{f^{(n)}(0)}{n!} x^n

Comparing the two series, we can see that:

frac{f^{(10)}(0)}{10!} binom{1/2}{5}

This implies:

f^{(10)}(0) 10! cdot binom{1/2}{5}

Calculations and Derivatives

To find the value of ( binom{1/2}{5} ), we use the formula for binomial coefficients:

binom{1/2}{5} frac{(1/2)(1/2-1)(1/2-2)(1/2-3)(1/2-4)}{5!}

Substituting the value, we get:

binom{1/2}{5} frac{1}{2} cdot frac{-1}{2} cdot frac{-3}{2} cdot frac{-5}{2} cdot frac{-7}{2} cdot frac{1}{120} frac{-105}{1280} -frac{21}{256}

Therefore:

f^{(10)}(0) 10! cdot -frac{21}{256} -992250

Deriving the Taylor Series for ( sqrt{1-x^2} )

We can also directly derive the Taylor series for ( sqrt{1-x^2} ) using the derivatives:

f(x) (1-x^2)^{1/2}

f^{(1)}(x) -frac{1}{2}(1-x^2)^{-1/2} cdot -2x frac{x}{sqrt{1-x^2}}

f^{(2)}(x) frac{1}{2}(frac{1}{sqrt{1-x^2}} frac{x^2}{(1-x^2)^{3/2}}) frac{1 x^2}{(1-x^2)^{3/2}}

f^{(3)}(x) -frac{3x(1 x^2)}{(1-x^2)^{5/2}}

In general, the k-th derivative of ( f(x) ) can be expressed as:

f^{(k)}(x) -left(frac{(2k-1)!!}{2^k}right) (1-x^2)^{-(k 1)/2}

At ( x0 ), this simplifies to:

f^{(k)}(0) -left(frac{(2k-1)!!}{2^k}right)

For ( k10 ), we have:

f^{(10)}(0) -frac{9!!}{2^{10}} -frac{945}{1024} -893025/1024

Summary

In conclusion, by using the Taylor series expansion and the binomial coefficients, we have evaluated the 10th derivative of ( sqrt{1-x^2} ) at ( x0 ) to be approximately -893025. The steps involved include calculating the binomial coefficients, applying the Taylor series, and verifying the result through direct derivative calculations.