/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 55 Find \(\frac{f(x+h)-f(x)}{h}\) a... [FREE SOLUTION] | 91影视

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Chapter 14: Problem 55

Find \(\frac{f(x+h)-f(x)}{h}\) and simplify. $$f(x)=x^{4}+7$$

Short Answer

Expert verified
The derivative of the function \(f(x)=x^{4}+7\) using the limit definition is \(4x^{3}\).

Step by step solution

01

Identify the Function f(x+h)

First, replace all occurrences of \(x\) with \(x+h\) in the function. The function becomes \(f(x+h)=(x+h)^{4}+7\). Apply the binomial theorem to expand \((x+h)^{4}\).
02

Simplify the Function

Now, subtract \(f(x)\) from \(f(x+h)\) to form the function \(\frac{f(x+h)-f(x)}{h}\). This simplifies to \(\frac{(x+h)^{4}+7-(x^{4}+7)}{h}\). We can cancel out '+7' from both the numerator. Therefore, the expression simplifies to \(\frac{(x+h)^{4}-x^{4}}{h}\).
03

Expand and Simplify further

Next, expand \((x+h)^{4}\) to \(x^{4} + 4x^{3}h + 6x^{2}h^{2} + 4xh^{3}+ h^{4}\). Substitute this back into the function. The expression simplifies to \(\frac{4x^{3}h + 6x^{2}h^{2} + 4xh^{3}+ h^{4}}{h}\). Since every term in the numerator has a factor of \(h\), each term can be simplified by dividing by \(h\).
04

Take the limit

Take the limit of the function as \(h\) approaches zero, \(lim_{h->0}\) {. After substituting \(h\) with 0, the expression simplifies to \(4x^{3}\).

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