What are the derivative rules needed for these functions?

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In summary, we discussed the application of various differentiation rules (chain rule, product rule, quotient rule, power rule) to different functions, including trigonometric and rational functions. We also corrected a mistake in the solution provided for part f).
  • #1
Joyci116
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Homework Statement



a.) f(x)=tan2(x)

b.) cos3(x2)

c.) (2x-1)/(5x+2)

d.) (sqrt(x2-2x))(secx)

e.) f(x)=((2x+3)/(x+7))3/2

f.) [sin(x)cos(x)]2

Homework Equations


chain rule
Product rule
Quotient rule
Power rule



The Attempt at a Solution


a.) would you do the power rule for this? 2tanx
b.) this is a combination of the chain rule and the power rule?
-3sinx2*2x
c.) use the quotient rule
((5x+2)(2)-(2x-1)(5))/(5x+2)2

((10x+2)-(10x-5))/(5x+2)2

7/(5x+2)2

d.) use the chain rule and the product rule?
Use the chain rule for the first pararenthasis. And then use the product rule?
f.) used the chain rule
2sinxcosx*(-cosxsinx)
 
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  • #2
a) this is actually both chain and product rule. tan[itex]^{2}[/itex]x is the same as (tanx)[itex]^{2}[/itex].
So now you use power rule on the entire function, multiplied by the derivative of the function, i.e. 2tanxsec[itex]^{2}[/itex]x

b) Again, chain rule and power rule. cos[itex]^{3}[/itex](x[itex]^{2}[/itex]) can be rewritten as (cos(x[itex]^{2}[/itex]))[itex]^{3}[/itex], which, when differentiated, becomes
3(cos(x[itex]^{2}[/itex]))[itex]^{2}[/itex](-sin(x[itex]^{2}[/itex]))(2x)

c) Looks right

d) yes

e) Combination quotient rule / power rule / chain rule. first differentiate as if it were a single variable, then differentiate what's inside using quotient rule.

f) the first part looks right, 2sinxcosx, but the 2nd part doesn't. The 2nd part should basically be (d/dx)(sinxcosx) which is product rule, i.e. cos[itex]^{2}[/itex]x - sin[itex]^{2}[/itex]x
 
  • #3
thank you.
 
  • #4
PShooter1337 said:
a) this is actually both chain and product rule. tan[itex]^{2}[/itex]x is the same as (tanx)[itex]^{2}[/itex].
So now you use power rule on the entire function, multiplied by the derivative of the function, i.e. 2tanxsec[itex]^{2}[/itex]x

b) Again, chain rule and power rule. cos[itex]^{3}[/itex](x[itex]^{2}[/itex]) can be rewritten as (cos(x[itex]^{2}[/itex]))[itex]^{3}[/itex], which, when differentiated, becomes
3(cos(x[itex]^{2}[/itex]))[itex]^{2}[/itex](-sin(x[itex]^{2}[/itex]))(2x)

c) Looks right

d) yes

e) Combination quotient rule / power rule / chain rule. first differentiate as if it were a single variable, then differentiate what's inside using quotient rule.

f) the first part looks right, 2sinxcosx, but the 2nd part doesn't. The 2nd part should basically be (d/dx)(sinxcosx) which is product rule, i.e. cos[itex]^{2}[/itex]x - sin[itex]^{2}[/itex]x

Don't provide solutions here in the future. It violates the PF rules that you agreed to when you joined here.
 

FAQ: What are the derivative rules needed for these functions?

What are derivatives and why are they important in science?

Derivatives are mathematical tools used to describe the rate of change of a function. They are important in science because they allow us to understand how a system changes over time or in response to different variables.

How do you find derivatives?

There are several methods for finding derivatives, including the power rule, product rule, quotient rule, and chain rule. These methods involve manipulating the original function to find the slope of a tangent line at a specific point.

Why is it important to find more derivatives?

Finding more derivatives allows us to better understand the behavior of a function or system. It can also help us make predictions about future changes and identify patterns in the data.

What are some real-world applications of finding derivatives?

Derivatives are used in a variety of fields, including physics, engineering, economics, and biology. They can be used to model population growth, predict the motion of objects, optimize production processes, and much more.

How can I improve my ability to find derivatives?

Practice is key when it comes to finding derivatives. Start with simple functions and work your way up to more complex ones. It's also helpful to understand the different rules and techniques for finding derivatives and to familiarize yourself with common functions and their derivatives.

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