# Find the derivative of the following function

• meeklobraca
The correct use of the chain rule for this problem is 3Cos^2(5x^2-6) * -sin(5x^2-6) * 10x. In summary, to find the derivative of cos^3(5x^2-6), you must use the chain rule twice, resulting in the final answer of -30xCos^2(5x^2-6)Sin(5x^2-6).
meeklobraca

## Homework Statement

y = cos^3(5x^2-6)

## The Attempt at a Solution

I used the chain rule for this question but I am not sure about the cos^3 part. Using the chain rule i got

-10xSin^3(5x^2-6)

But was I supposed to use the power rule on the cos^3 part? I can think of 3 ways this could be done but I am not sure which way to go about it.

Thanks!

You have to think of both cosine AND the exponent 3 as external functions. So you need to perform the chain rule twice:

(d/dx) cos^3(5x^2-6) = 3 (cos(5x^2-6))^2 * (d/dx) cos(5x^2-6), at which you point you'll also need to do the chain rule cos(5x^2-6).

How do I do the chain rule twice with this?

I got 3 Cos(5x-6)^2 by -sin (5x^2-6)?

Do I do something like ( 3 (-sin (5x^2-6) d/dx (5x^2-6))^2 ) (-sin (5x^2-6)

?

I think I was making this too complicated.

=d/dx Cos^3 d/dx (5x^2-6)
=3Cos^2(5x^2-6) -sin(5x^2-6) 10x
=-30xCos^2(5x^2-6)Sin(5x^2-6)

What do you guys an gals think?

That is correct.

## 1. What is a derivative?

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

## 2. Why do we need to find the derivative of a function?

Finding the derivative of a function allows us to understand the behavior of the function at a given point. It can help us determine the maximum and minimum points, the direction of the function's change, and the concavity of the curve.

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

To find the derivative of a function, we use the rules of differentiation, which include the power rule, product rule, quotient rule, and chain rule. These rules allow us to manipulate the function and find its derivative.

## 4. What is the difference between the derivative and the derivative of a function?

The derivative is a concept that represents the rate of change of any function. The derivative of a specific function is the function that results from finding the derivative using the rules of differentiation.

## 5. What are some real-life applications of finding the derivative of a function?

The concept of derivatives is used in various fields such as physics, engineering, economics, and finance. For example, derivatives can be used to find the velocity and acceleration of an object in motion, optimize production processes in engineering, and predict changes in stock prices in finance.

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