Derivatives and integrals help

In summary: As for \displaystyle\frac{d}{dt}(t^5)\,, you can use the power rule.In summary, to find the derivative of the integral over e^t to t^5 (sqrt(8+x^4)) dx, you need to use the chain rule and the power rule. The result will be F'(t^5) multiplied by the derivative of t^5, and F'(x) is equal to the square root of 8 plus x to the power of 4.
  • #1
polulech
4
0
derivative of integral over e^t to t^5 (sqrt(8+x^4)) dx

I know I need to use the chain rule and I can take the derivative of the integral without respect to e^t and t^5. If you know the answer, can you answer and tell me how to do it?! Calculus final on Monday...
 
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  • #2
polulech said:
derivative of integral over e^t to t^5 (sqrt(8+x^4)) dx

I know I need to use the chain rule and I can take the derivative of the integral without respect to e^t and t^5. If you know the answer, can you answer and tell me how to do it?! Calculus final on Monday...

Have you never seen the formula [tex] \frac{d}{dt}\int_{A(t)}^{B(t)} f(x,t) dx =
\frac{dB(t)}{dt} \left. f(x,t)\right|_{x=B(t)}
- \frac{dA(t)}{dt} \left. f(x,t)\right|_{x=A(t)}
+ \int_{A(t)}^{B(t)} \frac{\partial f(x,t)}{\partial t} \, dx ? [/tex]

RGV
 
  • #3
Umm no I have not :(
How do I put the pieces from my integral into that?
 
  • #4
polulech said:
derivative of integral over e^t to t^5 (sqrt(8+x^4)) dx

I know I need to use the chain rule and I can take the derivative of the integral without respect to e^t and t^5. If you know the answer, can you answer and tell me how to do it?! Calculus final on Monday...

Hi polulech. Welcome to PF.

Suppose F(x) is the anti-derivative of [itex]\displaystyle \sqrt{8+x^4\,}\,.[/itex]

Then by the fundamental theorem of calculus, [itex]\displaystyle \int_a^b\sqrt{8+x^4\,}\,dx=F(b)-F(a)\,.[/itex]

In the case of your integral you have: [itex]\displaystyle \int_{e^t}^{\,t^5}\sqrt{8+x^4\,} \,dx=F(t^5)-F(e^t)\,.[/itex]

You know that [itex]\displaystyle \frac{d}{dx}F(x)=\sqrt{8+x^4\,}\,. [/itex] Combine this result with the chain rule to find the derivative of your integral.
 
  • #5
how would I do that
 
  • #6
polulech said:
how would I do that

Can you find [itex]\displaystyle \frac{d}{dt}F(t^5)\,,[/itex] if you know that F'(x)=√(8 + x4) ?
 
  • #7
would i plug t^5 into the √8+x^4 and then calculate the derivative?
 
  • #8
polulech said:
would i plug t^5 into the √8+x^4 and then calculate the derivative?
No.

F(x) is a function whose derivative is √(8+x4).

The chain rule says that [itex]\displaystyle\frac{d}{dt}F(t^5)=F\,'(t^5)\cdot \frac{d}{dt}(t^5)\,.[/itex]

Furthermore, [itex]\displaystyle F\,'(t^5)=\sqrt{8+t^{20}}[/itex]
 

What are derivatives and integrals?

Derivatives and integrals are mathematical operations used to calculate the rate of change of a function and the area under a curve, respectively. They are fundamental concepts in calculus and are used to solve a variety of problems in science and engineering.

Why are derivatives and integrals important?

Derivatives and integrals are important because they provide a way to analyze and understand the behavior of complex systems. They allow us to model and predict the behavior of physical phenomena, such as the motion of objects, the growth of populations, and the flow of fluids.

How are derivatives and integrals related?

Derivatives and integrals are related through the fundamental theorem of calculus. This theorem states that the derivative of an integral is equal to the original function, and the integral of a derivative is equal to the original function plus a constant. In other words, derivatives and integrals are inverse operations of each other.

What are some real-world applications of derivatives and integrals?

Derivatives and integrals have countless real-world applications, including predicting stock market trends, optimizing manufacturing processes, and modeling climate change. They are also used in fields such as physics, engineering, economics, and biology.

How can I improve my understanding of derivatives and integrals?

To improve your understanding of derivatives and integrals, it is important to practice solving problems and applying these concepts to real-world scenarios. You can also seek help from a tutor or join a study group to gain a deeper understanding of the subject. Additionally, there are many online resources and textbooks available to help you learn and master derivatives and integrals.

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