How to differentiate y=cos4(x2 + ex)?

In summary, the problem is to differentiate the function y=cos4(x2+ex) and the attempted solution is to use the substitution u=x^2+e^x along with the chain rule. However, this doesn't work because the cos4 part is a separate function.
  • #1
futurept
13
0

Homework Statement


The problem is to differentiate the function.
y=cos4(x2 + ex)


Homework Equations


cos(x)'= -sin(x), (xn)'=n*xn-1, (ex)'=ex*x'


The Attempt at a Solution


Thought it would be the chain rule. Here's what I came up with:

y'=4cos3(-sin(x2 + ex)(2x + ex)

Is this right? If not, any suggestions on what I did wrong?
 
Last edited:
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  • #2
futurept said:
y'=4cos3(-sin(x2 + ex)(2x + ex)

Is this right? If not, any suggestions on what I did wrong?

Is this a typo? What are you taking the cosine of?
 
  • #3
no it's not a typo. my answer was y'= 4cos3(-sin(x2 + ex))*(2x + ex)
 
  • #4
I did forget to say that the problem says not to simplify.
 
  • #5
You're close but take another look at the chain rule.

f(x) = h(g(x))

f'(x) = h'(g(x)) * g'(x)

Your answer follows the following incorrect differentiation.

f'(x) = h'(g'(x))

But you only messed up for one part of the chain. The innermost function was differentiated correctly.
 
  • #6
futurept said:
no it's not a typo. my answer was y'= 4cos3(-sin(x2 + ex))*(2x + ex)

In that case it is incorrect.

Try using the substitution [itex]u=x^2+e^x[/itex] along with the chain rule...What is [tex]\frac{d}{dx}\cos^4(u)[/tex]?
 
  • #7
so h(x)=cos4(x2+ex)

and g(x)= x2 + ex

but would the cos4 part be another function by itself? So then it would be f(g(h(x))).

and by definition of the chain rule it would be f '(g(h(x)))*g'(h(x))*h'(x)?
 
  • #8
futurept said:
so h(x)=cos4(x2+ex)

and g(x)= x2 + ex

but would the cos4 part be another function by itself? So then it would be f(g(h(x))).

and by definition of the chain rule it would be f '(g(h(x)))*g'(h(x))*h'(x)?
hmmm this doesn't make much sense.

Instead, let's call [itex]g(x) \equiv \cos^4(x)[/itex] and [itex]h(x)\equiv x^2+e^x[/itex] then [itex]f(x)\equiv\cos^4(x^2+e^x)=g(h(x))[/itex]

...follow?

Now, what does the chain rule give you for [itex]f'(x)[/itex]?
 
  • #9
yeah, i think i follow. so from your definition, f'(x)=g'(h(x))*h'(x)?
 
  • #10
so, (cos(x^2+e^x))^4

f'(x)= 4cos(x^2+e^x)^3*-sin(x^2+e^x)*(2x+e^x)
 
  • #11
futurept said:
so, (cos(x^2+e^x))^4

f'(x)= 4cos(x^2+e^x)^3*-sin(x^2+e^x)*(2x+e^x)

Yes, much better!:approve:
 
  • #12
you guys are awesome
 

What is differentiation?

Differentiation is a mathematical process used to find the rate of change of a function with respect to its independent variable. It involves finding the derivative of a function, which represents the slope of the function at a specific point.

Why is differentiation important in science?

Differentiation is important in science because it helps us understand how a system or process changes over time. For example, in physics, differentiation is used to find the velocity and acceleration of an object. In chemistry, it is used to determine reaction rates. In biology, it is used to study growth and development. In general, differentiation helps us analyze and make predictions about complex systems.

What is the difference between differentiation and integration?

Differentiation and integration are inverse operations. Differentiation finds the rate of change of a function, while integration finds the total or accumulated change of a function. In other words, differentiation is like zooming in on a function to find its slope, while integration is like zooming out to find the area under the curve.

How do you differentiate a function?

To differentiate a function, you follow a set of rules based on the properties of derivatives. The most common rules are the power rule, product rule, quotient rule, and chain rule. These rules involve taking the derivative of each term in the function and applying algebraic operations to simplify the result.

What are some real-world applications of differentiation?

Differentiation is used in a variety of fields, including physics, chemistry, biology, economics, and engineering. Some specific applications include finding optimal solutions in optimization problems, analyzing data in scientific research, and predicting future trends in financial markets.

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