If g(x)=x sin^-1(x / 4)+√(16-x^2), find g^'(2).

Name: Problem 33, Chapter 3: Essential Calculus Early Transcendentals
Uploaded: 2020-06-10T00:36:07-08:00
Duration: 1 min 50 s
Channel: Ernest Castorena
Description: If g(x)=x sin^-1(x / 4)+√(16-x^2), find g^'(2).

Key Concepts

Product Rule

The product rule is a fundamental differentiation technique used when taking the derivative of a function that is the product of two or more functions. It states that the derivative of a product is the derivative of the first function multiplied by the second function plus the first function multiplied by the derivative of the second function. This rule is essential when dealing with terms where variables are multiplied by other functions of the variable.

Chain Rule

The chain rule is a critical method for differentiating composite functions, where one function is applied to the result of another function. It allows you to compute the derivative of complex functions by taking the derivative of the outer function and then multiplying it by the derivative of the inner function. This technique is widely used in calculus to break down complicated differentiation problems into simpler parts.

Derivative of Inverse Trigonometric Functions

Differentiating inverse trigonometric functions, such as the inverse sine function, requires knowing their specific derivative formulas. For example, the derivative of the inverse sine function involves a radical expression that accounts for the geometry of the sine function. Mastery of these derivative formulas is crucial when dealing with functions that include inverse trigonometric components.

Differentiation of Radical Functions

When differentiating functions that include radical expressions, such as square roots, it is often necessary to apply the chain rule to handle the inner function under the radical. The differentiation process typically results in a quotient where the denominator involves the radical. Understanding how to differentiate radicals is important in addressing a wide range of problems in calculus.

Evaluation of Derivative at a Point

Evaluating the derivative at a specific point involves first finding the general derivative of the function using the appropriate rules and then substituting the given value of the variable into this derivative. This process determines the instantaneous rate of change of the function at that particular point, which is a core concept in differential calculus.

Transcript

Please give Ace some feedback