Question

Differentiate the function. $$u=\sqrt[5]{t}+4 \sqrt{t^{5}}$$

Name: Problem 23, Chapter 3: Calculus
Uploaded: 2020-03-25T23:27:02-08:00
Duration: 37 s
Channel: Amrita Bhasin
Description: Differentiate the function. u=√(t)+4 √(t^5)

   Differentiate the function.
$$u=\sqrt[5]{t}+4 \sqrt{t^{5}}$$

Calculus

James Stewart 4th Edition

Chapter 3, Problem 23

Instant Answer

Solved by Expert Amrita Bhasin

03/25/2020

Step 1

First, let's find the derivative of $\sqrt[5]{t}$. Using the power rule for differentiation, we have: $$\frac{d}{dt} \left(\sqrt[5]{t}\right) = \frac{1}{5} \cdot t^{\frac{1}{5} - 1} = \frac{1}{5} \cdot t^{-\frac{4}{5}}$$ Show more…

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Differentiate the function. $$u=\sqrt[5]{t}+4 \sqrt{t^{5}}$$

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Key Concepts

Power Rule

The Power Rule is a fundamental technique in calculus for finding the derivative of a function in the form of t^n. It states that the derivative of t raised to any power n is n*t^(n-1), making it essential for differentiating polynomial expressions and functions expressed with fractional exponents.

Fractional Exponents

Fractional exponents provide an alternative notation for expressing root functions, such as converting the fifth root or square root into t^(1/5) or t^(1/2) respectively. This conversion simplifies the process of differentiation since standard rules like the Power Rule can then be directly applied.

Constant Multiple Rule

The Constant Multiple Rule indicates that when a function is multiplied by a constant, the derivative of the product is the constant times the derivative of the function. This rule works in conjunction with other differentiation rules to efficiently compute the derivative of scaled functions.

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