Compute the derivative of the function *𝑦*=4cos−1(*𝑥*5).

To compute the derivative of the function \( y = 4 \cos^{-1}\left(\frac{x}{5}\right) \), we can apply the chain rule. Let's go through the steps.

### Step 1: Differentiate the outer function

The derivative of \( \cos^{-1}(u) \) with respect to \( u \) is given by:
\[
\frac{d}{du} \cos^{-1}(u) = -\frac{1}{\sqrt{1 - u^2}}
\]

So for \( y = 4 \cos^{-1}(u) \), we have:
\[
\frac{dy}{du} = 4 \cdot \left(-\frac{1}{\sqrt{1 - u^2}}\right) = -\frac{4}{\sqrt{1 - u^2}}
\]

### Step 2: Apply the chain rule

In our case, \( u = \frac{x}{5} \). Now differentiate \( u \) with respect to \( x \):
\[
\frac{du}{dx} = \frac{1}{5}
\]

### Step 3: Combine the results

Using the chain rule:
\[
\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} = -\frac{4}{\sqrt{1 - \left(\frac{x}{5}\right)^2}} \cdot \frac{1}{5}
\]

### Step 4: Simplify

Simplifying the expression:
\[
\frac{dy}{dx} = -\frac{4}{5 \sqrt{1 - \frac{x^2}{25}}}
\]
\[
\frac{dy}{dx} = -\frac{4}{5 \sqrt{\frac{25 - x^2}{25}}}
\]
\[
\frac{dy}{dx} = -\frac{4}{\sqrt{25 - x^2}}
\]

### Final Answer:
\[
\frac{dy}{dx} = -\frac{4}{\sqrt{25 - x^2}}
\]

Would you like further clarification on any step?

Here are 5 related questions to explore further:
1. What is the domain of the function \( \cos^{-1}(x/5) \)?
2. How would the derivative change if the function was \( 4 \sin^{-1}(x/5) \)?
3. What is the second derivative of this function?
4. Can you compute the integral of \( 4 \cos^{-1}(x/5) \)?
5. How does the chain rule apply to other inverse trigonometric functions?

**Tip:** Always verify the domain of inverse trigonometric functions when differentiating them, as their domains restrict certain values of the input.

Compute the derivative of the function y = 4 cos⁻¹(x / 5).

Explore the step-by-step process of finding the derivative of y = 4 cos⁻¹(x / 5) with detailed application of the chain rule. This solution covers inverse trigonometric differentiation and simplifies the result to -4 / √(25 - x²). Ideal for students in Grades 11-12 or early college.

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