Questions: Use the chain rule to find the derivative of (f(x)=2 e^8 x^6-3 x^3) [f^prime(x)=]

Solution Steps

To solve this problem, we need to apply the "twisted" chain rule as given for the Z planet. The function \( f(x) = 2 e^{8x^6 - 3x^3} \) can be broken down into an outer function \( f(u) = 2e^u \) and an inner function \( u(x) = 8x^6 - 3x^3 \). We will first find the derivatives of these functions and then apply the twisted chain rule.

1. Identify the outer function \( f(u) = 2e^u \) and the inner function \( u(x) = 8x^6 - 3x^3 \).
2. Compute the derivative of the outer function \( f'(u) \).
3. Compute the derivative of the inner function \( u'(x) \).
4. Apply the twisted chain rule: \( f(g(x))' = \frac{f'(g)}{g'(x)} \).

Given the function \( f(x) = 2 e^{8x^6 - 3x^3} \), we can identify the outer function and the inner function for the chain rule.

• Outer function: \( h(u) = 2e^u \)
• Inner function: \( u(x) = 8x^6 - 3x^3 \)

The derivative of the outer function \( h(u) = 2e^u \) with respect to \( u \) is: \[ h'(u) = 2e^u \]

The derivative of the inner function \( u(x) = 8x^6 - 3x^3 \) with respect to \( x \) is: \[ u'(x) = \frac{d}{dx}(8x^6 - 3x^3) = 48x^5 - 9x^2 \]

On Z planet, the chain rule is given by: \[ f(g(x))' = \frac{f'(g)}{g'(x)} \] Here, \( f(g(x)) = h(u(x)) \).

Using the chain rule on Z planet: \[ f'(x) = \frac{h'(u)}{u'(x)} \]

Substitute \( h'(u) \) and \( u'(x) \) into the chain rule formula: \[ f'(x) = \frac{2e^{u}}{48x^5 - 9x^2} \]

Substitute \( u = 8x^6 - 3x^3 \) back into the expression: \[ f'(x) = \frac{2e^{8x^6 - 3x^3}}{48x^5 - 9x^2} \]

Final Answer