https://file.aiquickdraw.com/mathbot/file/1726073958474787hae92.jpg

The given problem asks to find \(\frac{dy}{dx}\) for the expression:

\[
y = (\sin^{-1} x)^x + (x)^{\cos^{-1} x}
\]

Let's break it down into two parts to differentiate:

### First Term: \((\sin^{-1} x)^x\)
Let \( u = (\sin^{-1} x)^x \).
We can apply logarithmic differentiation here.

1. Take the natural logarithm of both sides:
\[
\ln u = x \ln (\sin^{-1} x)
\]

2. Differentiate both sides with respect to \(x\) using the product rule:
\[
\frac{1}{u} \frac{du}{dx} = \ln (\sin^{-1} x) + x \cdot \frac{1}{\sin^{-1} x} \cdot \frac{1}{\sqrt{1 - x^2}}
\]

3. Therefore, the derivative of \(u\) is:
\[
\frac{du}{dx} = u \left( \ln (\sin^{-1} x) + \frac{x}{\sin^{-1} x \cdot \sqrt{1 - x^2}} \right)
\]
Now substitute \(u = (\sin^{-1} x)^x\) back into the equation.

### Second Term: \(x^{\cos^{-1} x}\)
Let \( v = x^{\cos^{-1} x} \).
Again, apply logarithmic differentiation.

1. Take the natural logarithm:
\[
\ln v = (\cos^{-1} x) \ln x
\]

2. Differentiate both sides using the product rule:
\[
\frac{1}{v} \frac{dv}{dx} = \frac{-1}{\sqrt{1 - x^2}} \ln x + \frac{\cos^{-1} x}{x}
\]

3. Therefore, the derivative of \(v\) is:
\[
\frac{dv}{dx} = v \left( \frac{-\ln x}{\sqrt{1 - x^2}} + \frac{\cos^{-1} x}{x} \right)
\]
Now substitute \(v = x^{\cos^{-1} x}\) back into the equation.

### Final Result
The total derivative \(\frac{dy}{dx}\) is:
\[
\frac{dy}{dx} = \frac{du}{dx} + \frac{dv}{dx}
\]

This gives the full derivative of the function. Would you like a detailed simplification of the expressions?

Here are five related questions to explore this further:

1. How does logarithmic differentiation simplify complex power functions?
2. Can you apply the product rule and chain rule to other trigonometric inverse functions?
3. What are some key applications of \(\sin^{-1} x\) and \(\cos^{-1} x\) in calculus?
4. How do we use logarithmic differentiation in real-life applications?
5. What are common mistakes to avoid when differentiating inverse trigonometric functions?

**Tip:** Always simplify complex expressions step-by-step when applying logarithmic differentiation to ensure clarity and correctness.

Find dy/dx if y = (sin^-1 x)^x + (x)^cos^-1 x.

This detailed solution shows how to find the derivative of y = (sin^-1 x)^x + x^cos^-1 x using logarithmic differentiation and the product rule. Step-by-step, learn the process of differentiating complex power functions involving inverse trigonometric expressions.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation