# Calculate Derivative of a function

• jweber
In summary, the first problem is asking for the derivative of the function g(z) = ((9z^2)/(6+z))^2. The suggested method is to use the quotient rule, and the derivative can be found using the formula (dy/dx)=f '(x)g(x) + f(x)g '(x). For the second problem, (dy/dx) needs to be calculated for the function y= (6x^2+x) (4x-x^2). The suggested methods are to either multiply the function out and then take the derivative, or to use the product rule. It is important to understand what a derivative is, which is the instantaneous slope on a graph at a specific point, and (
jweber
Calculate the derivative of the function

g(z) = ((9z^2)/(6+z))^2

Find g'(z) =

Also Calculate dy/dx of the following function

y= (6x^2+x) (4x-x^2)

You do not need to expand answer.

Any help is greatly appreciated thank you.

## The Attempt at a Solution

We don't do your homework for you, we help you with it. Now, what have you tried so far on the first one? Quotient rule seems to be the obvious choice...

Also Calculate dy/dx of the following function

y= (6x^2+x) (4x-x^2)

Ok, I can give you some hints with this one.

You have two options:

1. Multiply (6x^2+x) (4x-x^2) out then take the derivative.

2. Use product rule

Hint for product rule let f(x)=(6x^2+x) and g(x)=(4x-x^2) then use the following formula

(dy/dx)=f '(x)g(x) + f(x)g '(x)

The first option may sound like the easy way out but don't depend on it completely.

What if you had to find (dy/dx) of y=(x^2)(Ln(x))

Now you have no choice but to use the second option and the formula.

Also, I would like for you to actually know what a derivative is.

What is a derivative? You need to know this.

I'm going to say it right here so NEVER forget it.

A derivative is the instantaneous slope on a graph at a specific point.

There are other explanations but at least know that one.

We also need to talk about (dy/dx)

What does (dy/dx) mean?

You can read it as the derivative of y with respect to x.

(dy/dx) is actually an implicit differentiation.

And don't forget that you can also call (dy/dx) as y '

## What is the definition of a derivative?

A derivative is a mathematical concept that represents the rate of change of a function at a particular point. It can also be thought of as the slope of the tangent line to the function at that point.

## How do you calculate the derivative of a function?

The derivative of a function can be calculated using the formula: f'(x) = lim(h->0) [f(x+h) - f(x)] / h. This means taking the limit as the change in x (represented by h) approaches 0, and finding the difference in values of the function at that point and the neighboring point, divided by the change in x.

## What is the power rule for finding derivatives?

The power rule states that the derivative of a function with a single variable raised to a constant power is equal to the constant power multiplied by the original function with the exponent reduced by 1. For example, the derivative of x^3 is 3x^2.

## What is the chain rule and when is it used?

The chain rule is a method for finding the derivative of a composite function, where one function is nested within another. It states that the derivative of the outer function multiplied by the derivative of the inner function. It is typically used when finding the derivative of a function with multiple nested functions, such as sin(x^2).

## Why is it important to calculate derivatives?

Calculating derivatives is important because it allows us to analyze the rate of change of a function at a specific point. This information is useful in many fields, such as physics, economics, and engineering, where understanding how a system changes over time is crucial. Derivatives also play a key role in finding the maximum and minimum values of a function and solving optimization problems.

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