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Magazine Chapter 6: Problem 49
Find \(d y / d x\) $$ y=\sec ^{-1}\left(x^{3}\right) $$
Short Answer
Expert verified
\( \frac{dy}{dx} = \frac{3}{x\sqrt{x^6-1}} \) for \(x > 0\).
Step by step solution
01
Identify the function and recall the derivative of inverse secant
The given function is \( y = \sec^{-1}(x^3) \). We need to find the derivative \( \frac{dy}{dx} \). The derivative of \( \sec^{-1}(u) \) with respect to \( u \) is \( \frac{1}{|u|\sqrt{u^2-1}} \).
02
Determine the inner function u(x)
Rewrite the inside function of the inverse secant as \( u = x^3 \). This identifies \( u \) which will be used in the chain rule.
03
Differentiate the inner function u(x)
Differentiate \( u = x^3 \) with respect to \( x \). The derivative is \( \frac{du}{dx} = 3x^2 \).
04
Apply the Chain Rule
According to the chain rule, \( \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} \). Here, \( \frac{dy}{du} = \frac{1}{|x^3|\sqrt{(x^3)^2-1}} \) and \( \frac{du}{dx} = 3x^2 \).
05
Combine the results
Combine the results from steps 3 and 4 to find \( \frac{dy}{dx} = \frac{1}{|x^3|\sqrt{(x^3)^2-1}} \cdot 3x^2 \).
06
Simplify the expression
Simplify the expression \( \frac{dy}{dx} = \frac{3x^2}{|x^3|\sqrt{x^6-1}} \) by recognizing that \(|x^3| = x^3 \) for \(x > 0\) and \(-x^3\) for \(x < 0\). This leads to final expression \( \frac{3}{x\sqrt{x^6-1}} \) assuming \(x > 0\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Inverse Trigonometric Functions
Inverse trigonometric functions are essential in calculus for dealing with angles rather than lengths. One common example is the inverse secant function, denoted as \(\sec^{-1}(x)\). This function is the reverse of the secant function, and it returns the angle whose secant is \x\. The derivative of the inverse secant function with respect to \x\ is generally \( rac{1}{|x|\sqrt{x^2-1}}\). In problems involving inverse trigonometric functions, such as finding the derivative of \y = \sec^{-1}(x^3)\, we need to consider not only the derivative formula but also any internal functions, known as inner functions, which leads us to the chain rule.
Chain Rule
The chain rule is a powerful tool in calculus used for differentiating composite functions. If you have a function inside another function, the chain rule helps find the derivative. Essentially, it states: \(\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}\).
- Here, \(y\) is a function of \(u\), and \(u\) is a function of \(x\).
- This technique helps us by breaking down complicated expressions into simpler parts. For example, in \(y = \sec^{-1}(x^3)\), \(u=x^3\).
Derivative Calculation
Calculating derivatives involves applying rules and formulas systematically. In our exercise, we start by identifying \(y = \sec^{-1}(x^3)\) and recognize a need to differentiate the inner function \(u = x^3\). Differentiating gives \(\frac{du}{dx} = 3x^2\).For the inverse secant function, \(\frac{dy}{du}\) is given by \(\frac{1}{|u|\sqrt{u^2-1}}\). Substitute \(u\) back as \(x^3\) to get \\[ \frac{dy}{du} = \frac{1}{|x^3|\sqrt{(x^3)^2-1}}.\]Next, recall the chain rule: Multiply these two derivatives to find \(\frac{dy}{dx} = \frac{1}{|x^3|\sqrt{x^6-1}} \cdot 3x^2\). This results in a rational expression, setting the stage for further simplification.
Simplifying Expressions
Simplifying expressions requires careful attention to detail, ensuring each component is as simplified as possible. Here, the goal is to reduce \(\frac{3x^2}{|x^3|\sqrt{x^6-1}}\) to a more manageable form. We start by acknowledging that \( |x^3| = x^3 \) when \(x > 0\), which simplifies the expression. Therefore, for positive \(x\),\[ \frac{dy}{dx} = \frac{3}{x \sqrt{x^6-1}}.\]This final expression offers insight into the rate at which \(y\) changes concerning \(x\), providing a simplified, elegant solution perfect for further analysis or practical application in real-world problems.
