Calculus :( said:Homework Statement
I have the derived function:
f'(x) = [1/(1+kx)^2]e^[x/(1+kx)]
k is a positive constant
Homework Equations
I need to find the second derivative, which I thought was just the derivative of the exponent multiplied by the coefficient (as you find the first derivative), but the answers say:
f''(x) = [1/(1+kx)^4]e^[x/(1+kx)] + e^[x/(1+kx)] . [-2k/(1+kx)^3]
The Attempt at a Solution
All I managed to get was:
f''(x) = [1/(1+kx)^4]e^[x/(1+kx)]
##\frac{df}{dx}=\frac{e^{\frac{x}{1+kx}}}{(1+kx)^2}##. You have the quotient of two functions of ##x##. You can't just take the derivative of the exponential term. What do you do when you have a product or quotient of functions?Calculus :( said:Homework Statement
I have the derived function:
f'(x) = [1/(1+kx)^2]e^[x/(1+kx)]
k is a positive constant
Homework Equations
I need to find the second derivative, which I thought was just the derivative of the exponent multiplied by the coefficient (as you find the first derivative), but the answers say:
f''(x) = [1/(1+kx)^4]e^[x/(1+kx)] + e^[x/(1+kx)] . [-2k/(1+kx)^3]
The second derivative of an exponential function is the rate of change of the first derivative. It measures how quickly the slope of the original exponential function is changing at any given point.
To find the second derivative of an exponential function, you first need to take the derivative of the function using the rules of differentiation. Then, you take the derivative of that result to find the second derivative. This can be done using the chain rule or the quotient rule, depending on the form of the original function.
The second derivative of an exponential function is important because it can provide information about the concavity and inflection points of the original function. It can also help in determining the maximum and minimum values of the function.
Yes, the second derivative of an exponential function can be negative. This indicates that the slope of the original function is decreasing at that point. It can also mean that the function is concave down at that point.
The second derivative of an exponential function is related to the original function in that it describes the rate of change of the first derivative. It can also provide information about the shape and behavior of the original function, such as its concavity or inflection points.