Find the derivative of the function

In summary, the speaker is struggling with finding the derivative of two functions in a calculus class and is looking for help. They mention that they just learned the chain rule and would like to use that to solve the problems, but are open to other methods as well. They provide the answers for both problems and thank the listener for their assistance. A summary of the solution for the first problem is also provided.
  • #1
bjon-07
84
0
I am in math 50 (calc one) and i would greatly appreate it if someone could please show me how to solve these problems. I have the anwsers, but i can't figure out how to get them. thank you

In both problems i am need to find the derivative of the function,(we just learned the chain rule, so i am 'supposed to' use that to figure them out. But if you know a another way, great.

problem one

y=te^(-t^2) i normally know how to solve this problems but the t before the e is messing me up. the answer is e^(-t^2)(1-2t^2)

second problem

y=(z^(1/2)/e^z) the answer is e^-z/(2(z^(1/2))-(z^(1/2))e^-Z

thank you again for your help.
 
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  • #2
Originally posted by bjon-07
In both problems i am need to find the derivative of the function,(we just learned the chain rule, so i am 'supposed to' use that to figure them out. But if you know a another way, great.

problem one

y=te^(-t^2) i normally know how to solve this problems but the t before the e is messing me up. the answer is e^(-t^2)(1-2t^2)


The following two formulae are useful

1. d/dx ex = ex

2. d/dx eg(x)= g'(x) eg(x) (this is an example of chain rule, if you need further explanation, please tell.)

I'll do the first question and the second one is similar to the first one.

Let
f(t) = te(-t2)

f'(t) = t d/dt(e(-t2)) + e(-t2)d/dt (t)

= t*(-2t)* e(-t2) (use formula 2 above) + e(-t2)

= e-t2[1-2t2]
 
  • #3
K L Kam did a good job so I'll just content my self with pointing out, since bjon-07 said specifically that it was the "t before the e" that was giving him trouble: use the PRODUCT rule!
(fg)'= f g'+ f' g

KL Kam used it when he said
"f'(t) = t d/dt(e(-t2)) + e(-t2)d/dt (t)"
 
Last edited by a moderator:
  • #4
Thank you for your help

Thank you for your help. I was trying to solve the whole thing using only the chain rule ( i did not use the product rule).
 

1. What is the definition of a derivative?

A derivative is a mathematical concept that represents the rate of change of a function at a specific point. It is defined as the slope of the tangent line to the function at that point.

2. How do you find the derivative of a function?

To find the derivative of a function, you can use the power rule, product rule, quotient rule, or chain rule, depending on the type of function. These rules involve manipulating the function algebraically and using the limit concept to find the rate of change at a specific point.

3. What is the purpose of finding the derivative of a function?

The derivative of a function has many applications in mathematics and science. It can be used to find the maximum and minimum values of a function, determine the slope of a curve, and analyze the behavior of a function at a certain point. It also has practical applications in fields such as economics, physics, and engineering.

4. Can you find the derivative of any function?

In most cases, yes. However, there are some functions that do not have a derivative, such as those with discontinuities or sharp corners. These functions are not differentiable and therefore do not have a defined rate of change at certain points.

5. How can the derivative of a function be represented graphically?

The derivative of a function can be represented graphically as the slope of the tangent line at each point on the function's graph. It can also be represented as the rate of change over time, as the slope of a curve on a velocity-time or acceleration-time graph. Additionally, the derivative can be shown as the area under the curve of a function's graph on a rate-time graph.

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