Chapter 2: Q 1 (page 187)

Use the $z \to x$ definition of the derivative to show that $\frac{d}{d x} (x^{4}) = 4 x^{3}$ .

Short Answer

Expert verified

$\lim_{z \to x} \frac{z^{4} - x^{4}}{z - x} = \lim_{z \to x} \frac{(z - x) (z^{3} + x z^{2} + x^{2} z + x^{3})}{z - x} = \lim_{z \to x} (z^{3} + x z^{2} + x^{2} z + x^{3}) = x^{3} + x \times x^{2} + x^{2} \times x + x^{3} = x^{3} + x^{3} + x^{3} + x^{3} = 4 x^{3}$

Step by step solution

Step 1. Given Information

We have given the following function :-

$x^{4}$ .

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

$\frac{d}{d x} (x^{4}) = 4 x^{3}$

Step 2. Derivative of given function :-

Consider the given function as :-

$f (x) = x^{4}$ .

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 :-

localid="1648379528748" $\frac{d}{d x} (x^{4}) = \lim_{z \to x} \frac{z^{4} - x^{4}}{z - x}$

Simplify it :-

localid="1648379632597" $\frac{d}{d x} (x^{4}) = \lim_{z \to x} \frac{z^{4} - x^{4}}{z - x} \Rightarrow \frac{d}{d x} (x^{4}) = \lim_{z \to x} \frac{(z - x) (z^{3} + x z^{2} + x^{2} z + x^{3})}{z - x} \Rightarrow \frac{d}{d x} (x^{4}) = \lim_{z \to x} (z^{3} + x z^{2} + x^{2} z + x^{3}) \Rightarrow \frac{d}{d x} (x^{4}) = x^{3} + x \times x^{2} + x^{2} \times x + x^{3} \Rightarrow \frac{d}{d x} (x^{4}) = x^{3} + x^{3} + x^{3} + x^{3} \Rightarrow \frac{d}{d x} (x^{4}) = 4 x^{3}$

Hence proved.

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Use the $z \to x$ definition of the derivative to show that $\frac{d}{d x} (x^{4}) = 4 x^{3}$ .

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Derivative of given function :-

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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

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Use the $z \to x$ definition of the derivative to show that $\frac{d}{d x} (x^{4}) = 4 x^{3}$ .