step 1 Okay, so we have the function x to the power of one -fifth, cosine of x square. We are trying to

F(x)=x^1 / 5 cosx^2

Name: Problem 6, Chapter 10: Fundamentals of Differential Equations
Uploaded: 2019-08-15T14:39:03-08:00
Duration: 1 min 30 s
Channel: Arnab Bose
Description: f(x)=x^1 / 5 cosx^2

Key Concepts

Product Rule

The product rule is a fundamental differentiation technique used when dealing with the derivative of the product of two functions. It states that the derivative of a product is the derivative of the first function multiplied by the second plus the first function multiplied by the derivative of the second, which is essential when a function is expressed as a product of two or more sub-functions.

Chain Rule

The chain rule is a critical tool in differentiation for dealing with composite functions. It allows you to calculate the derivative of a function in which one function is nested inside another by differentiating the outer function with respect to the inner function and then multiplying by the derivative of the inner function.

Power Rule

The power rule simplifies the process of differentiating functions of the form x^n, where n is any real number, including fractions. It states that the derivative of x^n is n*x^(n-1), making it indispensable for functions featuring exponentiation.

Fractional Exponents

Working with fractional exponents involves understanding functions where the exponent is a fraction, requiring careful handling of domain and differentiability issues. This concept is important as it extends the idea of exponentiation to non-integer powers, which has broad applications in calculus.

Composite Functions

Composite functions involve applying one function to the result of another, creating nested expressions. Differentiating such functions typically requires the use of the chain rule, highlighting the creativity and interconnection between different differentiation techniques.

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