Questions: Differentiate. f(x)=x^9 e^x/(x^9+e^x) f^prime(x)= x

Solution Steps

To differentiate the given function \( f(x) = \frac{x^9 e^x}{x^9 + e^x} \), we will use the quotient rule. The quotient rule states that if you have a function \( f(x) = \frac{g(x)}{h(x)} \), then its derivative is given by \( f'(x) = \frac{g'(x)h(x) - g(x)h'(x)}{(h(x))^2} \). Here, \( g(x) = x^9 e^x \) and \( h(x) = x^9 + e^x \). We will find the derivatives \( g'(x) \) and \( h'(x) \) using the product rule and basic differentiation rules, respectively, and then apply the quotient rule.

We are given the function \( f(x) = \frac{x^9 e^x}{x^9 + e^x} \). To find the derivative \( f'(x) \), we will use the quotient rule, which is given by:

\[ f'(x) = \frac{g'(x)h(x) - g(x)h'(x)}{(h(x))^2} \]

where \( g(x) = x^9 e^x \) and \( h(x) = x^9 + e^x \).

First, we differentiate the numerator \( g(x) = x^9 e^x \) using the product rule:

\[ g'(x) = \frac{d}{dx}(x^9 e^x) = x^9 \frac{d}{dx}(e^x) + e^x \frac{d}{dx}(x^9) = x^9 e^x + 9x^8 e^x \]

Next, we differentiate the denominator \( h(x) = x^9 + e^x \):

\[ h'(x) = \frac{d}{dx}(x^9 + e^x) = 9x^8 + e^x \]

Substitute \( g'(x) \), \( g(x) \), \( h'(x) \), and \( h(x) \) into the quotient rule formula:

\[ f'(x) = \frac{(x^9 e^x + 9x^8 e^x)(x^9 + e^x) - x^9 e^x (9x^8 + e^x)}{(x^9 + e^x)^2} \]

Simplify the expression for \( f'(x) \):

\[ f'(x) = \frac{(x^9 e^x + 9x^8 e^x)(x^9 + e^x) - x^9 e^x (9x^8 + e^x)}{(x^9 + e^x)^2} \]

This simplifies to:

\[ f'(x) = \frac{(x^9 + e^x)(x^9 e^x + 9x^8 e^x) - x^9 e^x (9x^8 + e^x)}{(x^9 + e^x)^2} \]

Final Answer