47-50 Find y^' and y " y=cos(x^2)

Name: Problem 47, Chapter 3: Calculus Early Transcendentals
Uploaded: 2020-10-15T16:17:59-08:00
Duration: 1 min 44 s
Channel: Darshan Maheshwari
Description: 47-50 Find y^' and y " y=cos(x^2)

Key Concepts

Chain Rule

The chain rule is a fundamental principle in calculus used to differentiate composite functions. In problems where a function is applied to another function, such as applying a cosine function to a quadratic function, the chain rule allows one to differentiate by taking the derivative of the outer function evaluated at the inner function and then multiplying by the derivative of the inner function.

Power Rule

The power rule is a basic differentiation rule that simplifies finding the derivative of a function expressed as a power of x. When the inner function is a polynomial like x², the power rule is applied to differentiate it by lowering the exponent by one and multiplying by the original exponent.

Differentiation of Trigonometric Functions

This concept involves understanding how to differentiate trigonometric functions such as cosine and sine. The derivative of the cosine function, for example, is -sine, and this knowledge is critical when dealing with functions where the cosine function is applied to another function, requiring the use of the chain rule.

Higher-Order Derivatives

Calculating higher-order derivatives, such as the second derivative, involves differentiating the first derivative. This process may require the application of additional differentiation rules, including the chain rule and the product rule, especially when the first derivative is itself a product or composition of functions, ensuring that all parts of the derivative are correctly accounted for.

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