Differentiate.
f(x) = 8 − x^ex/x + ex
f '(x) =
Hard to tell just what you mean there. If you mean
f(x) = (8-xe^x)/(x+e^x)
f' = [(-e^x - xe^x)(x+e^x) - (8-xe^x)(1+e^x)]/(x+e^x)^2
= (-xe^x - x^2 e^x - e^2x - xe^2x - 8 + xe^x - 8e^x + xe^2x)/(x+e^x)^2
= (e^x(-x+x^2-e^x-xe^x+x-8+xe^x)-8)/(x+e^x)^2
= (e^x(x^2-e^x-8)-8)/(x+e^x)^2
doesn't get much simpler
If I got the original wrong, fix it and visit wolframalpha.com
To differentiate the given function, we will use the power rule and the quotient rule of differentiation. Let's break down the process step by step:
Step 1: Apply the power rule to differentiate each term in the expression.
For the first term, 8, the derivative will be zero since it is a constant.
For the second term, x^ex, we will first differentiate the exponent x^ex and then multiply it by the derivative of the base, x. Let's break it down further:
- Differentiating the exponent, ex: This can be done using the chain rule. We can rewrite ex as e^(x*1). Now, using the chain rule, we differentiate the outer function (e^u) first (which gives e^u) and then multiply it by the derivative of the inner function (x), which is 1. So, the derivative of ex with respect to x is ex.
- Differentiating the base, x: The derivative of x with respect to x is simply 1.
So, the derivative of the second term, x^ex, is ex * 1 = ex.
For the third term, ex, the derivative with respect to x is ex.
Step 2: Apply the quotient rule to differentiate the entire expression f(x).
The quotient rule states that for two functions u(x) and v(x), the derivative of their quotient is given by:
(u'v - uv') / (v^2)
Here, the numerator will be: (8 * ex) - (x * ex) * 1, which simplifies to (8ex - xex).
The denominator will be: (x + ex)^2.
So, combining the numerator and denominator, the derivative of f(x) is:
f '(x) = (8ex - xex) / (x + ex)^2
Therefore, the derivative of f(x) is (8ex - xex) / (x + ex)^2.