Chapter 2: Q 2 (page 187)

Use the $z \to x$ definition of the derivative to show that role="math" localid="1648386605099" $\frac{d}{d x} (x^{8}) = 8 x^{7}$ .

Short Answer

Expert verified

$\lim_{z \to x} \frac{z^{8} - x^{8}}{z - x} = \lim_{z \to x} \frac{(z - x) (z^{7} + x z^{6} + x^{2} z^{5} + x^{3} z^{4} + x^{4} z^{3} + x^{5} z^{2} + x^{6} z + x^{7})}{z - x} = \lim_{z \to x} (z^{7} + x z^{6} + x^{2} z^{5} + x^{3} z^{4} + x^{4} z^{3} + x^{5} z^{2} + x^{6} z + x^{7}) = x^{7} + x^{7} + x^{7} + x^{7} + x^{7} + x^{7} + x^{7} + x^{7} = 8 x^{7} .$

Step by step solution

Step 1. Given Information

We have given the following function :-

$x^{8}$ .

We have to use the $z \to x$ definition of derivative, to prove that :-

$\frac{d}{d x} (x^{8}) = 8 x^{7}$

Step 2. Derivative of given function :-

Consider the given function as :-

$f (x) = x^{8}$ .

We know that $z \to x$ definition of derivative is stated that :-

$\frac{d}{d x} f (x) = \lim_{z \to x} \frac{f (z) - f (x)}{z - x}$

Put the values :-

role="math" localid="1648386886563" $\frac{d}{d x} (x^{8}) = \lim_{z \to x} \frac{z^{8} - x^{8}}{z - x}$

Simplify it :-

role="math" localid="1648386910268" $\frac{d}{d x} (x^{8}) \lim_{z \to x} \frac{z^{8} - x^{8}}{z - x} \Rightarrow \frac{d}{d x} (x^{8}) = \lim_{z \to x} \frac{(z - x) (z^{7} + x z^{6} + x^{2} z^{5} + x^{3} z^{4} + x^{4} z^{3} + x^{5} z^{2} + x^{6} z + x^{7})}{z - x} \Rightarrow \frac{d}{d x} (x^{8}) = \lim_{z \to x} (z^{7} + x z^{6} + x^{2} z^{5} + x^{3} z^{4} + x^{4} z^{3} + x^{5} z^{2} + x^{6} z + x^{7}) \Rightarrow \frac{d}{d x} (x^{8}) = x^{7} + x^{7} + x^{7} + x^{7} + x^{7} + x^{7} + x^{7} + x^{7} \Rightarrow \frac{d}{d x} (x^{8}) = 8 x^{7}$

Hence proved.

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Use the $z \to x$ definition of the derivative to show that role="math" localid="1648386605099" $\frac{d}{d x} (x^{8}) = 8 x^{7}$ .

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Derivative of given function :-

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Use the z→xdefinition of the derivative to show that role="math" localid="1648386605099" ddx(x8)=8x7.

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Derivative of given function :-

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Discrete Mathematics

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Study anywhere. Anytime. Across all devices.

Use the $z \to x$ definition of the derivative to show that role="math" localid="1648386605099" $\frac{d}{d x} (x^{8}) = 8 x^{7}$ .