复合函数fx=expsinxx那么fx的8阶导数在0处的绝对值是多少

我们可以使用链式法则来计算$f(x)$的8阶导数。首先，我们计算$f(x)$的一阶导数： $$f'(x) = \frac{d}{dx} \left( e^{\sin(x^2)} \right) = e^{\sin(x^2)} \cdot \frac{d}{dx} \left( \sin(x^2) \right) = e^{\sin(x^2)} \cdot \cos(x^2) \cdot \frac{d}{dx} \left( x^2 \right) = 2x \cdot e^{\sin(x^2)} \cdot \cos(x^2)$$

然后，我们计算$f(x)$的二阶导数： \begin{align*} f''(x) &= \frac{d}{dx} \left( 2x \cdot e^{\sin(x^2)} \cdot \cos(x^2) \right) \ &= 2 \cdot e^{\sin(x^2)} \cdot \cos(x^2) + 2x \cdot \frac{d}{dx} \left( e^{\sin(x^2)} \cdot \cos(x^2) \right) \ &= 2 \cdot e^{\sin(x^2)} \cdot \cos(x^2) + 2x \cdot \left( e^{\sin(x^2)} \cdot \frac{d}{dx} \left( \cos(x^2) \right) + \cos(x^2) \cdot \frac{d}{dx} \left( e^{\sin(x^2)} \right) \right) \ &= 2 \cdot e^{\sin(x^2)} \cdot \cos(x^2) + 2x \cdot \left( e^{\sin(x^2)} \cdot (-\sin(x^2)) \cdot \frac{d}{dx} \left( x^2 \right) + \cos(x^2) \cdot 2x \cdot e^{\sin(x^2)} \cdot \cos(x^2) \right) \ &= 2 \cdot e^{\sin(x^2)} \cdot \cos(x^2) - 2x^2 \cdot e^{\sin(x^2)} \cdot \cos^2(x^2) - 2x^2 \cdot e^{\sin(x^2)} \cdot \sin(x^2) \end{align*}

接下来，我们可以继续计算$f(x)$的三阶、四阶、五阶和六阶导数： \begin{align*} f'''(x) &= \frac{d}{dx} \left( 2 \cdot e^{\sin(x^2)} \cdot \cos(x^2) - 2x^2 \cdot e^{\sin(x^2)} \cdot \cos^2(x^2) - 2x^2 \cdot e^{\sin(x^2)} \cdot \sin(x^2) \right) \ &= 2 \cdot e^{\sin(x^2)} \cdot (-\sin(x^2)) \cdot \frac{d}{dx} \left( x^2 \right) - 4x \cdot e^{\sin(x^2)} \cdot \cos(x^2) + 4x^3 \cdot e^{\sin(x^2)} \cdot \cos^2(x^2) - 4x^3 \cdot e^{\sin(x^2)} \cdot \sin(x^2) \ &= -2x \cdot e^{\sin(x^2)} \cdot \sin(x^2) - 4x \cdot e^{\sin(x^2)} \cdot \cos(x^2) + 4x^3 \cdot e^{\sin(x^2)} \cdot \cos^2(x^2) - 4x^3 \cdot e^{\sin(x^2)} \cdot \sin(x^2) \end{align*}

\begin{align*} f^{(4)}(x) &= \frac{d}{dx} \left( -2x \cdot e^{\sin(x^2)} \cdot \sin(x^2) - 4x \cdot e^{\sin(x^2)} \cdot \cos(x^2) + 4x^3 \cdot e^{\sin(x^2)} \cdot \cos^2(x^2) - 4x^3 \cdot e^{\sin(x^2)} \cdot \sin(x^2) \right) \ &= -2 \cdot e^{\sin(x^2)} \cdot \sin(x^2) - 2x \cdot e^{\sin(x^2)} \cdot \cos(x^2) - 4 \cdot e^{\sin(x^2)} \cdot \cos(x^2) + 4x \cdot e^{\sin(x^2)} \cdot \sin(x^2) \ &+ 12x^2 \cdot e^{\sin(x^2)} \cdot \cos^2(x^2) - 8x^2 \cdot e^{\sin(x^2)} \cdot \cos(x^2) \cdot \sin(x^2) - 12x^2 \cdot e^{\sin(x^2)} \cdot \sin(x^2) + 4x^4 \cdot e^{\sin(x^2)} \cdot \cos^3(x^2) \ &- 12x^4 \cdot e^{\sin(x^2)} \cdot \cos^2(x^2) \cdot \sin(x^2) + 8x^4 \cdot e^{\sin(x^2)} \cdot \cos(x^2) \cdot \sin^2(x^2) + 12x^4 \cdot e^{\sin(x^2)} \cdot \cos(x^2) - 8x^4 \cdot e^{\sin(x^2)} \cdot \cos(x^2) \cdot \sin(x^2) \end{align*}

\begin{align*} f^{(5)}(x) &= \frac{d}{dx} \left( -2 \cdot e^{\sin(x^2)} \cdot \sin(x^2) - 2x \cdot e^{\sin(x^2)} \cdot \cos(x^2) - 4 \cdot e^{\sin(x^2)} \cdot \cos(x^2) + 4x \cdot e^{\sin(x^2)} \cdot \sin(x^2) \right. \ &\left. + 12x^2 \cdot e^{\sin(x^2)} \cdot \cos^2(x^2) - 8x^2 \cdot e^{\sin(x^2)} \cdot \cos(x^2) \cdot \sin(x^2) - 12x^2 \cdot e^{\sin(x^2)} \cdot \sin(x^2) + 4x^4 \cdot e^{\sin(x^2)} \cdot \cos^3(x^2) \right. \ &\left. - 12x^4 \cdot e^{\sin(x^2)} \cdot \cos^2(x^2) \cdot \sin(x^2) + 8x^4 \cdot e^{\sin(x^2)} \cdot \cos(x^2) \cdot \sin^2(x^2) + 12x^4 \cdot e^{\sin(x^2)} \cdot \cos(x^2) - 8x^4 \cdot e^{\sin(x^2)} \cdot \cos(x^2) \cdot \sin(x^2) \right) \ &= -2 \cdot e^{\sin(x^2)} \cdot \cos(x^2) - 2x \cdot e^{\sin(x^2)} \cdot \cos(x^2) - 2x \cdot e^{\sin(x^2)} \cdot (-\sin(x^2)) - 4 \cdot e^{\sin(x^2)} \cdot (-\sin(x^2)) \ &+ 4 \cdot e^{\sin(x^2)} \cdot \sin(x^2) + 4x \cdot e^{\sin(x^2)} \cdot \cos(x^2) + 12x^2 \cdot e^{\sin(x^2)} \cdot \cos^2(x^2) - 12x^3 \cdot e^{\sin(x^2)} \cdot \cos^2(x^2) \ &- 8x^3 \cdot e^{\sin(x^2)} \cdot \cos(x^2) \cdot \sin(x^2) - 12x^2 \cdot e^{\sin(x^2)} \cdot \cos(x^2) - 12x^2 \cdot e^{\sin(x^2)} \cdot (-\sin(x^2)) \ &+ 12x^3 \cdot e^{\sin(x^2)} \cdot \sin(x^2) + 4x^4 \cdot e^{\sin(x^2)} \cdot 3\cos^2(x^2) \cdot (-\sin(x^2)) - 24x^3 \cdot e^{\sin(x^2)} \cdot \cos(x^2) \cdot \sin(x^2) \ &+ 8x^4 \cdot e^{\sin(x^2)} \cdot \sin(x^2) + 8x^4 \cdot e^{\sin(x^2)} \cdot \cos(x^2) \cdot 2\sin(x^2) + 12x^4 \cdot e^{\sin(x^2)} \cdot \cos(x^2) \ &- 8x^4 \cdot e^{\sin(x^2)} \cdot \cos(x^2) \cdot \sin(x^2) - 8x^4 \cdot e^{\sin(x^2)} \cdot \cos(x^2) \cdot \sin(x^2) \end{align*}

\begin{align*} f^{(6)}(x) &= \frac{d}{dx} \left( -2 \cdot e^{\sin(x^2)} \cdot \cos(x^2) - 2x \cdot e^{\sin(x^2)} \cdot \cos(x^2) - 2x \cdot e^{\sin(x^2)} \cdot (-\sin(x^2)) - 4 \cdot e^{\sin(x^2)} \cdot (-\sin(x^2)) \right. \ &+ 4 \cdot e^{\sin(x^2)} \cdot \sin(x^2) + 4x \cdot e^{\sin(x^2)} \cdot \cos(x^2) + 12x^2 \cdot e^{\sin(x^2)} \cdot \cos^2(x^2) - 12x^3 \cdot e^{\sin(x^2)} \cdot \cos^2(x^2) \ &- 8x^3 \cdot e^{\sin(x^2)} \cdot \cos(x^2) \cdot \sin(x^2) - 12x^2 \cdot e^{\sin(x^2)} \cdot \cos(x^2) - 12x^2 \cdot e^{\sin(x^2)} \cdot (-\sin(x^2)) \ &+ 12x^3 \cdot e^{\sin(x^2)} \cdot \sin(x^2) + 4x^4 \cdot e^{\sin(x^2)} \cdot 3\cos^2(x^2) \cdot (-\sin(x^2)) - 24x^3 \cdot e^{\sin(x^2)} \cdot \cos(x^2) \cdot \sin(x^2) \ &+ 8x^4 \cdot e^{\sin(x^2)} \cdot \sin(x^2) + 8x^4 \cdot e^{\sin(x^2)} \cdot \cos(x^2) \cdot 2\sin(x^2) + 12x^4 \cdot e^{\sin(x^2)} \cdot \cos(x^2) \ &- 8x^4 \cdot e^{\sin(x^2)} \cdot \cos(x^2) \cdot \sin(x^2) - 8x^4 \cdot e^{\sin(x^2)} \cdot \cos(x^2) \cdot \sin(x^2) \left. \right)' \ &= 2 \cdot e^{\sin(x^2)} \cdot \sin(x^2) - 2 \cdot e^{\sin(x^2)} \cdot \cos(x^2) - 2x \cdot e^{\sin(x^2)} \cdot \sin(x^2) - 2x \cdot e^{\sin(x^2)} \cdot \cos(x^2) \ &- 2x \cdot e^{\sin(x^2)} \cdot \cos(x^2) + 4 \cdot e^{\sin(x^2)} \cdot \sin(x^2) + 4 \cdot e^{\sin(x^2)} \cdot \cos(x^2) + 4x \cdot e^{\sin(x^2)} \cdot \sin(x^2) \ &+ 12x^2 \cdot e^{\sin(x^2)} \cdot (-2\cos(x^2)) \cdot \sin(x^2) - 24x^2 \cdot e^{\sin(x^2)} \cdot \cos(x^2) \cdot \sin(x^2) \ &+ 24x^3 \cdot e^{\sin(x^2)} \cdot \cos(x^2) \cdot \sin(x^2) + 12x^2 \cdot e^{\sin(x^2)} \cdot \sin(x^2) - 12x^3 \cdot e^{\sin(x^2)} \cdot \sin(x^2) \ &+ 4x^4 \cdot e^{\sin(x^2)} \cdot 3\cos^2(x^2) \cdot (-\sin(x^2)) + 4x^4 \cdot e^{\sin(x^2)} \cdot 3\cos^2(x^2) \cdot \cos(x^2) - 8x^3 \cdot e^{\sin(x^2)} \cdot \cos(x^2) \cdot \sin(x^2) \ &- 8x^3 \cdot e^{\sin(x^2)} \cdot \cos(x^2) \cdot \sin(x^2) + 8x^4 \cdot e^{\sin(x^2)} \cdot \sin(x^2) + 8x^4 \cdot e^{\sin(x^2)} \cdot \cos(x^2) \cdot 2\sin(x^2) \ &+ 16x^4 \cdot e^{\sin(x^2)} \cdot \cos^2(x^2) + 12x^4 \cdot e^{\sin(x^2)} \cdot \cos(x^2) - 8x^4 \cdot e^{\sin(x^2)} \cdot \cos(x^2) \cdot \sin(x^2) \ &- 8x^4 \cdot e^{\sin(x^2)} \cdot \cos(x^2) \cdot \sin(x^2) - 8x^4 \cdot e^{\sin(x^2)} \cdot \cos(x^2) \cdot \sin(x^2) \end{align*}

现在，我们可以继续计算$f(x)$的七阶和八阶导数： \begin{align*} f^{(7)}(x) &= \frac{d}{dx} \left( 2 \cdot e^{\sin(x^2)} \cdot \sin(x^2) - 2 \cdot e^{\sin(x^2)} \cdot \cos(x^2) - 2x \cdot e^{\sin(x^2)} \cdot \sin(x^2) - 2x \cdot e^{\sin(x^2)} \cdot \cos(x^2) \right. \ &- 2x \cdot e^{\sin(x^2)} \cdot \cos(x^2) + 4 \cdot e^{\sin(x^2)} \cdot \sin(x^2) + 4 \cdot e^{\sin(x^2)} \cdot \cos(x^2) + 4x \cdot e^{\sin(x^2)} \cdot \sin(x^2) \ &+ 12x^2 \cdot e^{\sin(x^2)} \cdot (-2\cos(x^2)) \cdot \sin(x^2) - 24x^2 \cdot e^{\sin(x^2)} \cdot \cos(x^2) \cdot \sin(x^2) \ &+ 24x^3 \cdot e^{\sin(x^2)} \cdot \cos(x^2) \cdot \sin(x^2) + 12x^2 \cdot e^{\sin(x^2)} \cdot \sin(x^2) - 12x^3 \cdot e^{\sin(x^2)} \cdot \sin(x^2) \ &+ 4x^4 \cdot e^{\sin(x^2)} \cdot 3\cos^2(x^2) \cdot (-\sin(x^2)) + 4x^4 \cdot e^{\sin(x^2)} \cdot 3\cos^2(x^2) \cdot \cos(x^2) - 8x^3 \cdot e^{\sin(x^2)} \cdot \cos(x^2) \cdot \sin(x^2) \ &- 8x^3 \cdot e^{\sin(x^2)} \cdot \cos(x^2) \cdot \sin(x^2) + 8x^4 \cdot e^{\sin(x^2)} \cdot \sin(x^2) + 8x^4 \cdot e^{\sin(x^2)} \cdot \cos(x^2) \cdot 2\sin(x^2) \ &+ 16x^4 \cdot e^{\sin(x^2)} \cdot \cos^2(x