Derivative from First Principles for 3x-\frac{5}{x}

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In summary, the conversation is about finding the derivative of 3x-\frac{5}{x} from first principles. The derivative is found by taking the limit as h approaches 0 of \frac{3h-\frac{5}{x+h}+\frac{5}{x}}{h}, which simplifies to \frac{-5x+5(x+h)}{(x+h)hx}. The conversation ends with the speaker asking for help in simplifying the derivative.
  • #1
Checkfate
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Hello. I have been trying to find this derivative from first principles for at least a couple hours, but just can't make any progress with it.

Find the derivative of [tex]3x-\frac{5}{x}[/tex]

Well I start by saying that the derivative is the limit as h approaches 0 of

[tex]\frac{f(x+h)-f(x)}{h} [/tex] where h=deltax

Then I go on to say that this is equal to the limit as h approaches 0 of
[tex]\frac{3h-\frac{5}{x+h}+\frac{5}{x}}{h} [/tex]

I then simplify by taking h out of the numerator by factoring and then cancel h on the numerator and denominator. This the derivative equals the limit as h approches 0 of
[tex]3-\frac{5}{(x+h)h}+\frac{5}{xh} [/tex]

As you can see, my derivative is now a bloody mess and I see no way of getting h out of the denominator. Please help! By the way I need to do this from first principles (dy/dx=(f(x+h)-f(x))/(h) ). Thankyou!
 
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  • #2
Checkfate said:
Then I go on to say that this is equal to the limit as h approaches 0 of
[tex]\frac{3h-\frac{5}{x+h}+\frac{5}{h}}{h} [/tex]

The last term in the numrator is [tex]\frac{5}{x}[/tex].
 
  • #3
try combining the fractions into:
[tex]\frac{-5x+5(x+h)}{(x+h)hx} [/tex]
EDIT: can't seem to get that latex to work... anyone see where I went wrong with it?
 
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1. What is a derivative?

A derivative is a mathematical concept that describes the rate of change of a function at a specific point. It represents the slope of the tangent line to the function at that point.

2. Why do we need to find derivatives?

Derivatives are useful in many areas of science and engineering, such as physics, economics, and engineering. They can help us understand the behavior of a system, optimize processes, and make predictions.

3. How do you find a derivative?

The most common method for finding a derivative is using the rules of differentiation, such as the power rule, product rule, quotient rule, and chain rule. These rules allow us to find the derivative of a function by manipulating its algebraic form.

4. What is the difference between a derivative and an antiderivative?

A derivative gives us the slope of a function at a specific point, while an antiderivative reverses the process and gives us the original function. In other words, a derivative tells us how a function changes, while an antiderivative tells us the function itself.

5. Can you find the derivative of any function?

In theory, yes, the derivative of any function can be found using the rules of differentiation. However, in practice, some functions may be too complex to differentiate algebraically, and numerical methods may be required to find the derivative.

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