D/dx[sin^(-1)(x)]

Definition

The derivative of the inverse sine function, represented as $$\frac{d}{dx}[sin^{-1}(x)]$$, provides the rate of change of the angle whose sine is x. This derivative is crucial for understanding how the inverse sine function behaves, particularly in terms of its range and domain, which are essential features when dealing with inverse trigonometric functions.

5 Must Know Facts For Your Next Test

1. The derivative $$\frac{d}{dx}[sin^{-1}(x)]$$ is equal to $$\frac{1}{\sqrt{1 - x^2}}$$ for $$-1 < x < 1$$.
2. The domain of $$sin^{-1}(x)$$ is restricted to values of x between -1 and 1, and its range is between $$-\frac{\pi}{2}$$ and $$\frac{\pi}{2}$$.
3. This derivative helps find slopes of tangents to the curve defined by $$y = sin^{-1}(x)$$, which can be visualized on a graph.
4. The result $$\frac{1}{\sqrt{1 - x^2}}$$ indicates that as x approaches -1 or 1, the slope tends towards infinity, showing vertical tangents at these endpoints.
5. Understanding this derivative is essential when solving problems involving rates of change in contexts such as physics or engineering.

Review Questions

• How do you apply the chain rule to find the derivative of a composition involving sin^(-1)(x)?
  • To apply the chain rule for finding the derivative of a composition with $$sin^{-1}(x)$$, first identify the outer and inner functions. In this case, if you have a function like $$sin^{-1}(g(x))$$, you would differentiate it as $$\frac{d}{dx}[sin^{-1}(g(x))] = \frac{g'(x)}{\sqrt{1 - (g(x))^2}}$$. This highlights how the chain rule helps determine how changes in g(x) affect the overall function.
• What significance does the domain and range of sin^(-1)(x) hold when calculating its derivative?
  • The domain and range of $$sin^{-1}(x)$$ are crucial when calculating its derivative because they define where the function is valid and where its output exists. The domain being limited to $$-1 < x < 1$$ ensures that we only take derivatives where the inverse sine is defined. This also directly impacts the behavior of its derivative; for example, outside this domain, the derivative does not exist, reflecting how understanding these constraints helps us interpret results accurately.
• Evaluate how understanding d/dx[sin^(-1)(x)] can enhance problem-solving skills in calculus applications.
  • Understanding $$\frac{d}{dx}[sin^{-1}(x)]$$ allows you to tackle a variety of problems involving rates of change more effectively. By knowing that this derivative equals $$\frac{1}{\sqrt{1 - x^2}}$$, you can analyze scenarios in physics or engineering where angles play a role in motion or forces. This knowledge gives you tools to explore relationships between variables and predict outcomes based on angle changes. It not only enhances your ability to solve complex problems but also builds a deeper appreciation for how trigonometric concepts apply in real-world situations.

Related terms

Inverse Trigonometric Functions: Functions that reverse the action of the standard trigonometric functions, providing angles as outputs for given sine, cosine, or tangent values.

Derivative: A mathematical concept that represents the rate at which a function is changing at any given point, calculated as the limit of the average rate of change as the interval approaches zero.

Chain Rule: A fundamental rule in calculus used to differentiate compositions of functions, stating that the derivative of a composite function is the product of the derivative of the outer function and the derivative of the inner function.