Chapter 2: Q. 85 (page 199)

Use the $h \to 0$ definition of the derivative to prove the power rule holds for positive integers powers

Short Answer

Expert verified

We prove the power rule holds for positive integers powers using the definition of derivative

Step by step solution

Given information

We are given a function as $f (x) = x^{n}$ , Where n is a positive integer

Find the derivative

We use definition of the derivative

We have,

$\lim_{h \to 0} \frac{f (x + h) - f (x)}{h} \lim_{h \to 0} \frac{{(x + h)}^{n} - x^{n}}{h} \lim_{h \to 0} \frac{x^{n} + n x^{n - 1} h + . . . . . + h^{n} - x^{n}}{h} [b i n o m i a l e x p a n s i o n] \lim_{h \to 0} \frac{n x^{n - 1} h + . . . . . + h^{n}}{h} \lim_{h \to 0} \frac{h (n x^{n - 1} + . . . . + h^{n - 1)}}{h} \lim_{h \to 0} (n x^{n - 1} + . . . . + h^{n - 1)} = n x^{n - 1}$

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Use the $h \to 0$ definition of the derivative to prove the power rule holds for positive integers powers

Short Answer

Step by step solution

Given information

Find the derivative

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Use theh→0definition of the derivative to prove the power rule holds for positive integers powers

Short Answer

Step by step solution

Given information

Find the derivative

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Probability and Statistics

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Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.

Use the $h \to 0$ definition of the derivative to prove the power rule holds for positive integers powers