Differentiate the exponential function

In summary, the conversation is about someone having trouble understanding how to find the gradients of the curve y = e^x when x = 0, 1, and -1. They know the answers are 1, 7.39, and 0.05, but are struggling with the process. The person they are talking to corrects their mistake of thinking the derivative of e^x is xe^(x-1) and reminds them that it is simply e^x.
  • #1
monkeyass
1
0
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
 
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  • #2
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!
 
  • #3
,

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!
 

Related to Differentiate the exponential function

What is the definition of an exponential function?

An exponential function is a mathematical function of the form f(x) = ab^x, where a is a constant and b is the base. The base is typically a positive number greater than 1, and the exponent x can be any real number.

How do you differentiate an exponential function?

To differentiate an exponential function, use the power rule for derivatives. First, bring the exponent down in front of the base, then subtract 1 from the exponent. So for the function f(x) = ab^x, the derivative is f'(x) = ab^x * ln(b).

What is the derivative of e^x?

The derivative of e^x is e^x. This is because the natural logarithm, ln, of e is equal to 1. So when differentiating f(x) = e^x using the power rule, the derivative is f'(x) = e^x * 1 = e^x.

How do you differentiate a function with a variable as an exponent?

You can differentiate a function with a variable as an exponent using the chain rule. First, take the derivative of the base function, then multiply it by the derivative of the exponent. So for the function f(x) = x^2, the derivative is f'(x) = 2x * ln(x).

Are there any special cases when differentiating exponential functions?

Yes, there are two special cases when differentiating exponential functions. The first is when the base is equal to 1. In this case, the derivative is always 0. The second case is when the base is equal to 0. In this case, the derivative is undefined.

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