What is the Definition of d(a^x)/dx?

In summary, the definition of a derivative for the function f(x)=a^x is calculated using the limit of (a^h-1)/h as h approaches 0. This limit can be simplified to f'(0), resulting in the equation f'(x) = f'(0)a^x.
  • #1
G01
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Hey. I'm having trouble understanding part of the definition of this derivative. Any help will be appreciated.

[tex] f(x)=a^x [/tex]

Using the definition of a derivative, the derivative of the above function is:

[tex]f'(x) = \lim_{h \rightarrow 0}\frac{a^{x+h} + a^x}{h} = [/tex]

[tex] \lim_{h \rightarrow 0} \frac{a^x(a^h - 1)}{h} [/tex]

Since a^x does not depend on h it can be taken outside the limit:

[tex] f'(x) = a^x \lim_{h \rightarrow 0} \frac{a^h-1}{h} [/tex]

Now here is where I get confused. The text tells me that:

[tex] \lim_{h \rightarrow 0} \frac{a^x(a^h - 1)}{h} = f'(0) [/tex] (1)

If that is true then [tex] f'(x) = f'(0)a^x [/tex], but I have no idea why equation 1 is the way it is? How is that limit equal to f'(0)?:confused:
 
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  • #2
G01 said:
Hey. I'm having trouble understanding part of the definition of this derivative. Any help will be appreciated.

[tex] f(x)=a^x [/tex]

Using the definition of a derivative, the derivative of the above function is:

[tex]f'(x) = \lim_{h \rightarrow 0}\frac{a^{x+h} + a^x}{h} = [/tex]

[tex] \lim_{h \rightarrow 0} \frac{a^x(a^h - 1)}{h} [/tex]

Since a^x does not depend on h it can be taken outside the limit:

[tex] f'(x) = a^x \lim_{h \rightarrow 0} \frac{a^h-1}{h} [/tex]

Now here is where I get confused. The text tells me that:

[tex] \lim_{h \rightarrow 0} \frac{a^x(a^h - 1)}{h} = f'(0) [/tex] (1)

If that is true then [tex] f'(x) = f'(0)a^x [/tex], but I have no idea why equation 1 is the way it is? How is that limit equal to f'(0)?:confused:

Note:

[tex] \lim_{h \rightarrow 0} \frac{(a^h - 1)}{h} = \lim_{h \rightarrow 0} \frac{(a^{(0 + h)} - a^0)}{h} = f'(0)[/tex]
 
  • #3
You've made some errors. In the second line, you should have a minus sign, not a plus sign in the numerator. Equation (1) should read:

[tex]\lim _{h \to 0}\frac{a^h - 1}{h} = f'(0)[/tex]

You already have the equation:

[tex]f'(x) = a^x\lim _{h \to 0}\frac{a^h - 1}{h}[/tex]

Substitute 0 for x, and recognize that [itex]a^0 = 1[/itex], and you'll see why the equation for f'(0) holds.
 
  • #4
Ahhh icic that was simpler than i thought. Thank you.
 

1. What is the definition of d(a^x)/dx?

The definition of d(a^x)/dx is the derivative of the function a^x with respect to x. In other words, it is the rate of change of a^x with respect to x.

2. How is d(a^x)/dx calculated?

The derivative of a^x can be calculated using the power rule, which states that the derivative of a function raised to a power is equal to the power multiplied by the function raised to the power minus one. In this case, the derivative of a^x is equal to ln(a) multiplied by a^x.

3. What is the significance of d(a^x)/dx in mathematics?

The derivative d(a^x)/dx is a fundamental concept in calculus and is used to analyze the rate of change of many functions, including exponential functions like a^x. It is also used in optimization problems, where finding the maximum or minimum of a function requires taking its derivative.

4. Can d(a^x)/dx be negative?

Yes, d(a^x)/dx can be negative. This means that the function a^x is decreasing at that point. The value of d(a^x)/dx is determined by the value of a and x, and can be either positive or negative depending on their values.

5. How does d(a^x)/dx change as x increases?

As x increases, d(a^x)/dx also increases. This is because the value of a^x is increasing at an exponential rate, resulting in a larger rate of change or derivative. Similarly, as x decreases, d(a^x)/dx decreases because the function a^x is decreasing at a slower rate.

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