Differentiation (Trig Function)

In summary, to find the second derivative of cos(x^2), you can use the product rule with -2x and sin(x^2), and then use the chain rule to find the derivative of sin(x^2). Combining these rules, the solution is -2xsin(x^2).
  • #1
SciGuy26
5
0

Homework Equations



Take the second derivative of:
cos(x^2)

The Attempt at a Solution



cos(x^2)

dy/du = -sin(u)
du/dx = 2x

dy/dx= -2xsin(x^2)

I don't know how to begin taking the derivative of this a second time...
 
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  • #2
What rule do you think you will need to use?
 
  • #3
Use the product rule with -2x and sin(x^2). You'll have to use the chain rule again when you calculate the derivative of sin(x^2).
 
  • #4
So I'd have 2xcos(u) + -2sin(u)

How am I to use the chain rule at this point?

Ugh I don't understand how combine rules with composite functions.
 
  • #5
nm...

Solution should be as follows:
-2xsin(x^2)

dy/du= sin(u)
du/dx= 2x

-2xsin(u) + -2xsin(U)

-2xcos(u) + -2sin(u)

= 2x(-2xcos(x^2) + -2sin(x^2))

Could someone please confirm? Thank you for the help.
 

What is differentiation?

Differentiation is a mathematical process used to find the rate of change or slope of a function at a specific point. It is also known as finding the derivative of a function.

What is the purpose of differentiation?

The purpose of differentiation is to help us understand the behavior of a function and how it changes over time or in response to varying inputs. It is also used to solve optimization problems and to find the maximum or minimum values of a function.

What is the difference between differentiation and integration?

Differentiation is the process of finding the rate of change of a function, while integration is the process of finding the area under a curve. In other words, differentiation helps us find the slope of a function, while integration helps us find the total change over a given interval.

What are the basic rules of differentiation?

The basic rules of differentiation include the power rule, product rule, quotient rule, and chain rule. These rules help us find the derivative of more complex functions by breaking them down into simpler parts and applying specific formulas.

How is differentiation used in real life?

Differentiation is used in a variety of fields, including physics, economics, engineering, and statistics. It is used to model and predict the behavior of systems, to optimize processes, and to analyze data. Some common applications include calculating speed and acceleration, finding the maximum profit for a business, and predicting the growth of a population over time.

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