Questions: Given f(x)=1/(6-x^2)^4, find f'(x).

Transcript text: Given $f(x)=\frac{1}{\left(6-x^{2}\right)^{4}}$, find $f^{\prime}(x)$.

Solution

Solution Steps

To find the derivative $ f'(x) $ of the function $ f(x) = \frac{1}{(6-x^2)^4} $, we can use the chain rule and the power rule. First, rewrite the function as $ f(x) = (6-x^2)^{-4} $. Then, apply the chain rule: differentiate the outer function and multiply by the derivative of the inner function.

Step 1: Rewrite the Function

The given function is $ f(x) = \frac{1}{(6-x^2)^4} $. We can rewrite this function using a negative exponent: \[ f(x) = (6-x^2)^{-4} \]

Step 2: Apply the Chain Rule

To find the derivative $ f'(x) $, we apply the chain rule. The chain rule states that if you have a composite function $ g(h(x)) $, then the derivative is $ g'(h(x)) \cdot h'(x) $.

Step 3: Differentiate the Outer Function

The outer function is $ g(u) = u^{-4} $, where $ u = 6-x^2 $. The derivative of $ g(u) $ with respect to $ u $ is: \[ g'(u) = -4u^{-5} \]

Step 4: Differentiate the Inner Function

The inner function is $ h(x) = 6-x^2 $. The derivative of $ h(x) $ with respect to $ x $ is: \[ h'(x) = -2x \]

Step 5: Combine Using the Chain Rule

Using the chain rule, the derivative $ f'(x) $ is: \[ f'(x) = g'(h(x)) \cdot h'(x) = -4(6-x^2)^{-5} \cdot (-2x) \]

Step 6: Simplify the Expression

Simplifying the expression, we get: \[ f'(x) = 8x(6-x^2)^{-5} \] \[ f'(x) = \frac{8x}{(6-x^2)^5} \]

Final Answer

\[ \boxed{f'(x) = \frac{8x}{(6-x^2)^5}} \]

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