Calculating Derivatives: How Many Different Ways Can You Do It?

In summary: Regarding the second attempt, I thought I could plug in the x-coordinate to get the tangent equation. Since when is that wrong?You have to use both the power rule and the chain rule for the derivative of (1+2x)^2. You get 2*(1+2x)*d(1+2x)/dx=4*(1+2x).Well 4*(1+2x) = 4+8x = 2+4x which is what I got.Since when is 4+8x=2+4x?fk378, I rearranged your OP a bit to make a point.I tried to solve it 4 different ways using the above
  • #1
fk378
367
0

Homework Statement


Find the tangent line and the normal line to the curve y=(1+2x)^2 at the point (1,9).

Homework Equations


f'(x)= [f(x+h) - f(x)] / h ->plug in x=1
f'(x)= [f(x+h) - f(x)] / h ->plug in (x+h)
Power Rule
Power Rule (after factoring out the polynomial)

The Attempt at a Solution


I tried to solve it 4 different ways using the above 4 different approaches:

Plugging in x=1 for the derivative formula yields 12
Plugging in (x+h) for the derivative formula yields 4+8x
Using the power rule for (1+2x)^2 = 2(1+2x) = 2+4x (simplified version of 4+8x)
Using the power rule after factoring out the polynomial is 1+4x+4x^2 = 4+8xRegarding the first attempt, I thought I could plug in the x-coordinate to get the tangent equation. Since when is that wrong?
 
Last edited:
Physics news on Phys.org
  • #2
You have to use both the power rule and the chain rule for the derivative of (1+2x)^2. You get 2*(1+2x)*d(1+2x)/dx=4*(1+2x).
 
  • #3
Well 4*(1+2x) = 4+8x = 2+4x which is what I got.
 
  • #4
Since when is 4+8x=2+4x?
 
  • #5
fk378, I rearranged your OP a bit to make a point.

fk378 said:
I tried to solve it 4 different ways using the above 4 different approaches:
Plugging in (x+h) for the derivative formula yields 4+8x
Using the power rule after factoring out the polynomial is 1+4x+4x^2 = 4+8x
Dick confirmed this is correct.
Using the power rule for (1+2x)^2 = 2(1+2x) = 2+4x
You forgot the factor of 2 when you take the derivative of 1+2x. With this factor of 2 you get the same result as above.
Plugging in x=1 for the derivative formula yields 12
Obviously this is not a global result; it is only valid at x=1. One question: What is 4+8x evaluated at x=1?
 
  • #6
4+8x simplified = 2+4x
 
  • #7
D H said:
Obviously this is not a global result; it is only valid at x=1. One question: What is 4+8x evaluated at x=1?


But isn't the question asking what the slope is at x=1? So why wouldn't plugging in work?
 
  • #8
fk378 said:
4+8x simplified = 2+4x

Yes. And 2 simplified is 1. So 2=1.
 
  • #9
fk378 said:
But isn't the question asking what the slope is at x=1? So why wouldn't plugging in work?

I ask again, what is 4+8x evaluated at x=1?
 
  • #10
Sorry D H,
yes it equals 1. But if it's asking for the equation of the tangent line, what do I regard 12 as?

Dick,
2 in the above equations is a common factor that can be factored out. 2=1 has no such common factor.
 
  • #11
How in the world is 4+8x evaluated at x=1 equal to 1? (Answer: 4+8x evaluated at x=1 is 12.) How in the world is 4+8x = 2+4x? (Answer: Its not.)

You appear have some very basic understanding issues here. I think you need some face-to-face time with your teacher.
 
  • #12
Oh I did mean 12. Sorry I was just thinking about the problem...
Is it not possible to simplify the equation? Does that just apply to derivatives?
 
  • #13
You can say 2*(2+4x) = 4+8x. You cannot say 2+4x=4+8x. They are different equations. You can't just throw that factor of two into the ozone layer.
 

What are derivatives?

Derivatives are financial instruments that derive their value from an underlying asset or security. They can be used for hedging, speculation, or arbitrage purposes.

What are the different types of derivatives?

There are three main types of derivatives: options, futures, and swaps. Options give the holder the right, but not the obligation, to buy or sell an underlying asset at a predetermined price. Futures contracts require the buyer to purchase an underlying asset at a specified price and date in the future. Swaps involve exchanging cash flows based on different variables, such as interest rates or currencies.

How are derivatives used in the financial industry?

Derivatives are used for various purposes in the financial industry. For example, they can be used to manage risk by hedging against potential losses, to speculate on market movements, or to diversify a portfolio.

What are the risks associated with derivatives?

Derivatives carry a high level of risk, as their value is based on the performance of an underlying asset or security. They can be complex financial instruments, and if not used properly, can result in significant losses. It is important to thoroughly understand the risks involved before investing in derivatives.

How are derivatives priced?

Derivatives are priced based on various factors, including the underlying asset's price, time until expiration, interest rates, volatility, and supply and demand. These factors are used to calculate the fair value of the derivative, which is then reflected in the market price.

Similar threads

  • Calculus and Beyond Homework Help
Replies
6
Views
793
  • Calculus and Beyond Homework Help
Replies
5
Views
947
  • Calculus and Beyond Homework Help
Replies
8
Views
346
  • Calculus and Beyond Homework Help
Replies
9
Views
1K
  • Calculus and Beyond Homework Help
Replies
2
Views
1K
  • Calculus and Beyond Homework Help
Replies
15
Views
1K
  • Calculus and Beyond Homework Help
Replies
4
Views
1K
  • Calculus and Beyond Homework Help
Replies
4
Views
1K
  • Calculus and Beyond Homework Help
Replies
3
Views
1K
  • Calculus and Beyond Homework Help
Replies
12
Views
2K
Back
Top