How do you find derivatives and what do they represent in calculus?

  • Thread starter Mol_Bolom
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In summary: I meant the 1st derivative of 15x^4+6x^2 is 60x^3+12xIn summary, the conversation was about derivatives and the process of finding them for different equations, including a quadratic. The correct derivatives were discussed and the concept of the differential was brought up. The conversation also touched on the general form of a differential and the different variables that can be used.
  • #1
Mol_Bolom
24
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I think I finally figured out derivitaves, but not sure...

The derivative/differential of x in
x is 1
3x + 5 is 3
5x^2 - 2 is 10

And breaking down a quadratic thus

3x^5 + 2x^3 + 5x - 1
The first derivative is 15x^4 + 6x^2
The second derivative is 60x^3 + 12x
Third 180x^2
Fourth 360x
And final 360.
 
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  • #2
Mol_Bolom said:
3x^5 + 2x^3 + 5x - 1
The first derivative is 15x^4 + 6x^2
The second derivative is 60x^3 + 12x
Third 180x^2
Fourth 360x
And final 360.

No, first and third are wrong.

1. 15x^4 + 6x^2 + 5
2. 60x^3 + 12x
3. 180x^2 + 12
4. 360x
5. 360
 
  • #3
The derivative of [tex]5x^2-2[/tex] is actually [tex]10x[/tex].
 
  • #4
Ah...That was the area I had a problem with...The final solitary number...
Wado (thanks)
 
  • #5
Mol_Bolom said:
I think I finally figured out derivitaves, but not sure...

The derivative/differential of x in
x is 1
3x + 5 is 3
5x^2 - 2 is 10
The derivative (not differential) of 5x2- 2 at at x= 1 is 10. More generally, the derivative of 5x2- 2 at any x is 10x. The differential of x is dx, the differential fo 3x+ 5 is 3dx and the differential of 5x2- 2 is 10x dx.

And breaking down a quadratic thus

3x^5 + 2x^3 + 5x - 1
The first derivative is 15x^4 + 6x^2
The first derivative is 15x4+ 6x2+ 5

The second derivative is 60x^3 + 12x
Third 180x^2
Fourth 360x
And final 360.
Well, I wouldn't call it "final". That's the fourth derivative. The fifth derivative, and all succeeding derivatives is 0.
 
  • #6
HallsofIvy said:
The derivative (not differential) of 5x2- 2 at at x= 1 is 10. More generally, the derivative of 5x2- 2 at any x is 10x. The differential of x is dx, the differential fo 3x+ 5 is 3dx and the differential of 5x2- 2 is 10x dx.

don't you mean the differential of y?

isn't the differential of x : dx?
 
  • #7
ice109 said:
don't you mean the differential of y?
I don't see any
y's in there. It can be any variable expressed as a function of x.
ice109 said:
isn't the differential of x : dx?
I think this is what Halls said also!
 
  • #8
woops
 

1. What is a derivative?

A derivative is a mathematical concept used to measure the rate of change of a function with respect to its independent variable. It represents the slope of the tangent line at a specific point on a curve.

2. Why are derivatives important?

Derivatives are important because they allow us to analyze the behavior of a function and make predictions about its future values. They also have a wide range of applications in fields such as physics, economics, and engineering.

3. How do you calculate a derivative?

To calculate a derivative, you can use the formula: f'(x) = lim(h->0) [f(x+h) - f(x)]/h. This is known as the limit definition of a derivative. Alternatively, you can use rules such as the power rule, product rule, and chain rule to find the derivative of a function.

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

A derivative measures the instantaneous rate of change of a function, while an integral measures the accumulated change of a function over a certain interval. In other words, a derivative tells us how fast a function is changing at a specific point, while an integral tells us the total change of a function over a range of values.

5. How can derivatives be used to optimize a function?

Derivatives can be used to find the maximum or minimum value of a function, which is useful for optimization problems. This involves finding the critical points of a function (where the derivative is equal to 0) and determining whether they correspond to a maximum or minimum value.

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