Find the slope of the tangent to the curve at the point where x = a

In summary, the conversation is about finding the derivative of a function using the fundamental definition of the derivative and the differentiation rule for powers. The discussion also includes suggestions and corrections regarding the equation used and the methods applied. The possibility of the OP's course not having encountered the power rule yet is also mentioned.
  • #1
Kaleb
49
0

Homework Statement


http://img214.imageshack.us/img214/4673/mathproblemnw5.png


Homework Equations


lim x->a[tex]\frac{f(a+h)-f(a)}{h}[/tex]


The Attempt at a Solution


Ive tried so many times to figure this out. I first substituted the equation into the formula above and multiplied by the conjugate and ended up removing h from the numerator and denominator leaving a -1/2[tex]\sqrt{a}[/tex]. As you can see I have already lost credit for the problem and cannot redo it, but I still want to know how to do it. Any tips and help is appreciated.
 
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  • #2
Um, why can't you just differentiate y directly without using the fundamental definition of the derivative? Why not just apply differentiation rule for powers?
 
  • #3
I would suggest you try writing that as -1/(2*sqrt(a)). I'm really hoping that's what you meant.
 
  • #4
Defennder said:
Um, why can't you just differentiate y directly without using the fundamental definition of the derivative? Why not just apply differentiation rule for powers?

It appears from the question type that it is quite possible the OP's course hasn't encountered the power rule yet. And it's still good to know how to do it both ways anyways =]

Kaleb - You probably just typo-ed, but your posted definition of the derivative is incorrect.
 
  • #5
If that is true then that is pretty odd. I thought schools always teach the mechanical aspects of calculus before introducing the fundamentals. I learned it that way.
 

1. What is the definition of slope of the tangent?

The slope of the tangent is the rate of change of a curve at a specific point. It represents the steepness of the curve at that point and is calculated by finding the limit of the slope of a secant line as the two points on the curve get closer and closer together.

2. How is the slope of the tangent calculated?

To calculate the slope of the tangent at a specific point on a curve, you can use the derivative function. This function takes the equation of the curve and evaluates it at the desired point, giving you the slope of the tangent at that point. Alternatively, you can also use the formula (y2-y1)/(x2-x1) where (x1,y1) and (x2,y2) are two points on the curve that are very close together.

3. Why is finding the slope of the tangent important?

The slope of the tangent is important because it helps us understand the behavior of a curve at a specific point. It can tell us if the curve is increasing or decreasing at that point and how steep the curve is at that point. This information is useful in many areas such as physics, engineering, and economics.

4. What is the significance of the slope of the tangent at x = a?

The slope of the tangent at x = a is significant because it represents the slope of the curve at a specific point. This information can be used to calculate the rate of change of the curve at that point, which can be useful in predicting future behavior of the curve or understanding its behavior in relation to other curves.

5. Can the slope of the tangent be negative?

Yes, the slope of the tangent can be negative. It can be positive, negative, or zero, depending on the behavior of the curve at that point. A negative slope indicates that the curve is decreasing at that point, while a positive slope indicates that the curve is increasing at that point.

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