To find the derivative of $f(x) = 7 \cos^{-1}(x^2)$, we will apply the chain rule. Here’s the step-by-step process:

Step 1: Recognize the outer and inner functions

The outer function is $7 \cos^{-1}(u)$, where $u = x^2$.

So, $f(x) = 7 \cdot \cos^{-1}(x^2)$.

Step 2: Derivative of the outer function

We know the derivative of $\cos^{-1}(u)$ with respect to $u$ is: $\frac{d}{du} \left( \cos^{-1}(u) \right) = -\frac{1}{\sqrt{1 - u^2}}$ Therefore, the derivative of the outer function (with respect to $u$) is: $\frac{d}{dx} \left( 7 \cdot \cos^{-1}(x^2) \right) = 7 \cdot \left( -\frac{1}{\sqrt{1 - (x^2)^2}} \right) = -\frac{7}{\sqrt{1 - x^4}}$

Step 3: Apply the chain rule for the inner function $u = x^2$

The derivative of $u = x^2$ with respect to $x$ is: $\frac{d}{dx} (x^2) = 2x$

Step 4: Combine the derivatives

Now, applying the chain rule, the derivative of $f(x)$ is: $f'(x) = \left( -\frac{7}{\sqrt{1 - x^4}} \right) \cdot 2x = -\frac{14x}{\sqrt{1 - x^4}}$

Final Answer:

$f'(x) = -\frac{14x}{\sqrt{1 - x^4}}$

Let me know if you want more details or have any questions!

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Related Questions:

1. How do you find the derivative of $\sin^{-1}(x)$?
2. Can you solve an example using the chain rule for a more complex function?
3. How would the derivative change if the function was $7 \cos^{-1}(x^3)$?
4. What is the domain of the function $f(x) = 7 \cos^{-1}(x^2)$?
5. How do implicit derivatives work for composite functions like this?

Tip:

When dealing with inverse trigonometric functions, always remember to apply the chain rule and carefully consider the derivative of the inverse trig function itself.