Differentiate the exponential function

[monkeyass]

Hi,

I am having a bit of trouble, i am getting ready for an exam, one the questions i have asks
"given the curve y = e^x, draw tangents to esitmate the gradients of the curve when
a)x=0, b)x=1, c)x=-1,
Now i know the answers are:
a) 1 b) 7.39 c)0.05
However the toruble i am having is understanding hwo to get to those points. For the first i got to this:
x= 0
dy/dx|x=0 =0e^(0-1), but then where do i go to get to the point where the answer become 1? Could someone help me with the others, i am having a large amoutn of difficulty to get this to click in my head, if anyone could help me out i would be very apreciable.

Monkey

 

[jdavel]

monkeyass,

You've either been studying way too hard or not nearly hard enough!

The derivative of e^x isn't xe^(x-1). You're thyinking of the derivative of x^a which is ax^(a-1). But those aren't the same thing.

The derivatve of e^x is e^x. Ring a bell?

Good luck on that exam!

 

[thepatientmental]

,

The exponential function is a type of mathematical function that is defined as f(x) = e^x, where e is a constant value approximately equal to 2.71828. This function is widely used in many fields of mathematics and science, including calculus, statistics, and physics. The main characteristic of the exponential function is its rapid growth, as the value of x increases, the value of e^x also increases at an exponential rate.

To differentiate the exponential function, we use the power rule of differentiation, which states that the derivative of x^n is nx^(n-1). In the case of the exponential function, we have e^x, which can be rewritten as e^(1*x). Using the power rule, we get the derivative of e^x as e^(1*x) * 1 = e^x.

Now, to find the gradients of the curve at specific points, we use the derivative of the function. In this case, we have f(x) = e^x, and we want to find the gradients at x=0, 1, and -1. To do this, we simply plug in the values of x into the derivative of the function, which is e^x, and we get the following:

a) x=0: f'(0) = e^0 = 1
b) x=1: f'(1) = e^1 = 2.71828
c) x=-1: f'(-1) = e^(-1) = 0.36788

These values represent the gradients of the curve at the specified points. To estimate the gradients, we can plot these points on the curve and draw tangents to them, as instructed in the question. This will give us a visual representation of the gradients at those points.

I hope this explanation helps you understand how to differentiate the exponential function and find the gradients at specific points. If you have any further questions, please feel free to ask. Good luck on your exam!