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Chapter 11: Problem 71

Write out the proof that \(\frac{d}{d x}\left(x^{4}\right)=4 x^{3}\).

Short Answer

Expert verified
Using the power rule for derivatives, we find the derivative of the function \(f(x) = x^4\): \(f'(x) = 4x^{4-1}\). Simplifying the expression, we obtain \(f'(x) = 4x^3\), which proves that the derivative of \(x^4\) is \(4x^3\).

Step by step solution

01

Identify the function

First, identify the given function, which is \(f(x) = x^4\). Our goal is to find the derivative of this function, \(f'(x)\).
02

Apply the power rule

According to the power rule, to find the derivative of a function in the form of \(x^n\), multiply the exponent by the variable and subtract one from the exponent. In this case, the exponent is 4. So, the power rule will give us: \[f'(x) = 4x^{4-1}\]
03

Simplify the result

Now, simplify the expression we found in Step 2: \[f'(x) = 4x^{3}\] The proof is now complete. The derivative of \(x^4\) with respect to x is indeed \(4x^3\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Calculus
Calculus is a branch of mathematics that deals with rates of change and the accumulation of quantities. It's a powerful tool for analyzing changes and understanding patterns in physical phenomena, economics, engineering, and beyond. At the heart of calculus lie two primary concepts: differentiation, which concerns the rate of change of functions, and integration, which is about the accumulation of quantities.

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