1.Find the derivative of the function.
f(x)=x/(e^x)
2. Find the derivative of the function.
f(x)=(e^x+e^(-x))/2
To find the derivatives of the given functions, we can use the rules of differentiation. The first rule we can utilize is the quotient rule.
1. Derivative of f(x) = x/(e^x):
To apply the quotient rule, we need to find the derivatives of both the numerator and denominator.
First, let's find the derivative of the numerator, which is x:
The derivative of x with respect to x is simply 1.
Next, let's find the derivative of the denominator, which is (e^x):
The derivative of e^x with respect to x is e^x.
Now, we can apply the quotient rule:
f'(x) = (derivative of numerator * denominator - numerator * derivative of denominator) / (denominator^2)
= (1 * e^x - x * e^x) / (e^x)^2
= (e^x - x * e^x) / e^(2x)
Therefore, the derivative of f(x) = x/(e^x) is f'(x) = (e^x - x * e^x) / e^(2x).
2. Derivative of f(x) = (e^x + e^(-x))/2:
To find the derivative, we need to apply the sum rule and the chain rule.
Let's start by applying the sum rule to the numerator, which is (e^x + e^(-x)):
The derivative of e^x with respect to x is e^x.
The derivative of e^(-x) with respect to x is -e^(-x).
Now, let's differentiate the denominator, which is 2:
The derivative of a constant (in this case, 2) is zero.
Applying the sum rule and the chain rule, we get:
f'(x) = [(e^x)' + (e^(-x))'] / 2
= (e^x - e^(-x)) / 2
Therefore, the derivative of f(x) = (e^x + e^(-x))/2 is f'(x) = (e^x - e^(-x)) / 2.