How To Find Arc Trig Derivatives? Stepbystep Guide

Finding derivatives of trigonometric functions, particularly arc trig functions, is a fundamental skill in calculus. The arc trig functions, also known as inverse trig functions, are crucial for solving equations involving trigonometric expressions. Here’s a step-by-step guide on how to find the derivatives of arc trig functions.

Introduction to Arc Trig Functions

Before diving into derivatives, let’s briefly review the arc trig functions:

• Arc Sine (arcsin(x)): The inverse function of sine. It returns the angle whose sine is the given value.
• Arc Cosine (arccos(x)): The inverse function of cosine. It returns the angle whose cosine is the given value.
• Arc Tangent (arctan(x)): The inverse function of tangent. It returns the angle whose tangent is the given value.

Derivatives of Arc Trig Functions

To find the derivatives of these functions, we use the following formulas:

1. Derivative of arcsin(x):

   • ( \frac{d}{dx} \arcsin(x) = \frac{1}{\sqrt{1-x^2}} )

2. Derivative of arccos(x):

   • ( \frac{d}{dx} \arccos(x) = -\frac{1}{\sqrt{1-x^2}} )

3. Derivative of arctan(x):

   • ( \frac{d}{dx} \arctan(x) = \frac{1}{1+x^2} )

These formulas are derived using the concept of implicit differentiation and the chain rule, starting from the definitions of the inverse trigonometric functions.

Step-by-Step Derivation Process

For a clearer understanding, let’s derive the derivative of arcsin(x) step by step, as an example. The process involves implicit differentiation:

1. Start with the Definition: ( y = \arcsin(x) ) implies ( \sin(y) = x ).

2. Differentiate Both Sides with Respect to x: Using the chain rule on the left side, we have ( \cos(y) \cdot \frac{dy}{dx} = 1 ).

3. Solve for ( \frac{dy}{dx} ): This gives us ( \frac{dy}{dx} = \frac{1}{\cos(y)} ).

4. Express ( \cos(y) ) in Terms of x: Since ( \sin(y) = x ), we know from the Pythagorean identity that ( \cos(y) = \sqrt{1 - \sin^2(y)} = \sqrt{1 - x^2} ).

5. Substitute Back: Thus, ( \frac{dy}{dx} = \frac{1}{\sqrt{1 - x^2}} ).

This process can be adapted for arccos(x) and arctan(x) with their respective starting equations.

Practical Applications

Understanding how to find the derivatives of arc trig functions is crucial in various applications, including physics, engineering, and computer science. For example, in physics, the trajectory of projectiles can be modeled using trigonometric functions, and their inverses are essential for calculating specific parameters like the angle of projection for a desired range.

Example Problems

To solidify your understanding, let’s consider a simple example:

• Find the derivative of ( \arcsin(2x) ).

Using the chain rule, where if ( y = \arcsin(u) ) and ( u = 2x ), we have:

• ( \frac{dy}{du} = \frac{1}{\sqrt{1-u^2}} )
• ( \frac{du}{dx} = 2 )

So, ( \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} = \frac{1}{\sqrt{1-(2x)^2}} \cdot 2 = \frac{2}{\sqrt{1-4x^2}} )

Conclusion

In conclusion, finding the derivatives of arc trig functions involves understanding and applying specific formulas derived from the definitions and properties of these functions. Practice and recognizing the patterns of these derivatives will make you proficient in handling a wide range of calculus problems involving arc trig functions. Remember, the key to mastering calculus is not just memorizing formulas but understanding the logical and mathematical reasoning behind them.

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