Chapter 2: Q. 68 (page 223)

Use the definition of the derivative and the definition of the number e to prove that $f (x) = e^{x}$ is its own derivative.

Short Answer

Expert verified

It is proved that derivative of the given function is $e^{x}$ only.

Step by step solution

Step 1. Given Information

The given function is $f (x) = e^{x}$

Step 2. Calculation

As we know that, from the definition of derivative, $\frac{d}{d x} (b^{x}) = b^{x} (\underset{h \to 0}{l i m} \frac{e^{h} - 1}{h} = 1) where \underset{h \to 0}{l i m} \frac{e^{h} - 1}{h} = 1$

Now, we will substitute $b = e$ then the function becomes,

$\frac{d}{d x} (e^{x}) = \underset{h \to 0}{l i m} \frac{e^{x + h} - e^{x}}{h} = e^{x} (\underset{h \to 0}{l i m} \frac{e^{h} - 1}{h}) = e^{x} (1) = e^{x}$

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Use the definition of the derivative and the definition of the number e to prove that $f (x) = e^{x}$ is its own derivative.

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Calculation

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Use the definition of the derivative and the definition of the number e to prove that f(x)=exis its own derivative.

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Calculation

One App. One Place for Learning.

Most popular questions from this chapter

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Use the definition of the derivative and the definition of the number e to prove that $f (x) = e^{x}$ is its own derivative.