Question

Differentiate the functions. $$y=x^{4}+4 \sqrt[4]{x}$$

   Differentiate the functions.
$$y=x^{4}+4 \sqrt[4]{x}$$

Calculus and Its Applications

Calculus and Its Applications

Larry J. Goldstein,… 14th Edition

Chapter 3, Problem 16

Instant Answer

Solved by Expert Stephen Hobbs

10/26/2020

Step 1

We can rewrite the square root as a power of 1/4. So, the function becomes: $$y=x^{4}+4x^{1/4}$$ Show more…

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Differentiate the functions. $$y=x^{4}+4 \sqrt[4]{x}$$

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Key Concepts

Differentiation

Differentiation is the process of finding the derivative of a function, which represents the rate at which the function's value changes with respect to changes in its input. It is a fundamental concept in calculus and is used to analyze and interpret the behavior of functions, including those that model real-world phenomena.

Power Rule

The power rule is a basic rule in calculus used to differentiate functions of the form x^n, where n is any real number. According to the power rule, the derivative of x^n is n*x^(n-1), making it particularly useful for differentiating polynomial functions and functions with exponents, including fractional exponents.

Fractional Exponents

Fractional exponents are a way of expressing roots using exponent notation, allowing radicals to be written in a more manipulable algebraic form. In calculus, rewriting radicals as fractional exponents enables the application of the power rule to differentiate functions that initially contain radical expressions.

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