# How do you find derivative equations?

• viet_jon
In summary, the conversation is about finding derivative equations and the use of differential calculus in solving them. The power rule and the derivative of a sum are mentioned as useful tools in finding the derivative of a function. The conversation also touches on the importance of understanding differential calculus and its application in higher education.
viet_jon
[SOLVED] how do you find derivative equations?

I'm not sure if this is the right sub forum.

## Homework Statement

f(x) = x^3 +6X^2 + 9x + 4

## The Attempt at a Solution

in my book, the first derrivative equation is 3x^2 + 12x + 9
and second equation as 6x + 12

how do you get to these equations?

I know how to calculate the difference... I'm guessing somehow that's how you get it.

I'm not even sure if my question makes sense. I'm working ahead of class, so there's a lot of gaps I need to fill in.

what you are actually looking at is 'differential calculus' and it isn't as simple as getting the difference or some such method. You should get a textbook on 'Differential Calculus'.

Do you know the derivative of $g(x)=a\,x^n$ ?

The rule you should use is the power rule
f(x) = x^a
f'(x) = ax^(a-1)
The derivative of a constant is 0.

So the derivative of x^3 is 3x^2.
Also, the derivative of the sum is the sum of the derivative. So, you can take the derivative of each part of the equation separately.
So x^3 + 6x^2 becomes 3x^2 + 12x.

tnkx for the quick reply...k...this is weird then...differential calculus? I'm taking a grade 12 class.

with the function in the first post...I'm suppose to find coordinates and nature of the critical points. And in the example, it says differentiate ... f'(x) = 3x^2 + 12x +9 = 0
without explaining how to I get that equation.

rainbow child...I have no clue what that means. That looks way ahead of me.angus...give me a few moments with that.

A critical point is where the first or second derivative is 0 or undefined. In the first derivative, the critical points will tell you where the original equation changes from increasing to decreasing. While the second derivative will tell you where the equation changes concavity.

AngusYoung93 said:
The rule you should use is the power rule
f(x) = x^a
f'(x) = ax^(a-1)
The derivative of a constant is 0.

So the derivative of x^3 is 3x^2.
Also, the derivative of the sum is the sum of the derivative. So, you can take the derivative of each part of the equation separately.
So x^3 + 6x^2 becomes 3x^2 + 12x.

thnkx...

i got it now...

If you haven't heard of differential calculus before but are attempted this, i would advise you to stop straight away. Learn from the foundations, get a good textbook on Differential Calculus.

I've heard of it, but from this forum only.

Differential Calculus is probably way ahead of me though. I don't understand why my grade 12 workbook required me to do it...and even worse, not explaining how it's derived.

I browsed through my brother's first year Uni calculus book, I have a better understanding of it now. The stuff looks really interesting btw... it would probably be more interesting if I understood it all.

viet_jon said:
I've heard of it, but from this forum only.

Differential Calculus is probably way ahead of me though. I don't understand why my grade 12 workbook required me to do it...and even worse, not explaining how it's derived.

I browsed through my brother's first year Uni calculus book, I have a better understanding of it now. The stuff looks really interesting btw... it would probably be more interesting if I understood it all.

i just don't understand how differential calculus is 'way ahead' of you if you are in 12th grade. I had differential calculus from my 11th grades beginning. Which education board does your school follow?

## 1. How do you find the derivative equation using the power rule?

The power rule states that the derivative of a function raised to a constant power is equal to the constant multiplied by the function raised to the power minus one. So, to find the derivative equation using the power rule, you would first multiply the constant by the function raised to the power minus one, and then subtract one from the power.

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

A derivative is a function that represents the instantaneous rate of change of another function at a specific point. An antiderivative, on the other hand, is a function that, when differentiated, gives the original function. In simpler terms, a derivative tells us how a function is changing, while an antiderivative tells us what function produced a given rate of change.

## 3. Can you find the derivative of a constant?

Yes, the derivative of a constant is always zero. This is because the derivative measures the rate of change of a function, and a constant has no rate of change. So, when you take the derivative of a constant, you end up with zero.

## 4. How do you find the derivative of a composite function?

To find the derivative of a composite function, you would use the chain rule. The chain rule states that the derivative of a composite function is equal to the derivative of the outer function evaluated at the inner function, multiplied by the derivative of the inner function. This can be written as f'(g(x)) * g'(x).

## 5. What is the role of derivatives in real-world applications?

Derivatives are used in many real-world applications, particularly in fields such as physics, engineering, and economics. They help us understand rates of change, optimize functions, and make predictions about future behavior. For example, in physics, derivatives are used to calculate velocity and acceleration, while in economics, they can be used to determine the optimal price for a product.

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