Question

Find the derivative of the function f(x)=(x⁴eˣ)/(x⁴eˣ) using the quotient rule:

Answer 1

To find the derivative of the function f(x) = (x⁴eˣ)/(x⁴eˣ) using the quotient rule, differentiate the numerator and denominator separately and apply the quotient rule formula.

Step-by-step explanation:

To find the derivative of the function f(x) = (x⁴eˣ)/(x⁴eˣ) using the quotient rule, we need to differentiate the numerator and denominator separately, and then apply the quotient rule formula. The quotient rule states that if we have a function f(x) = g(x)/h(x), where g(x) and h(x) are differentiable functions, then the derivative of f(x) is given by (h(x) * g'(x) - g(x) * h'(x))/(h(x))^2. Applying this to our function:

• Numerator: Differentiate x⁴eˣ using the product rule, and we get g'(x) = 4x³eˣ + x⁴eˣ
• Denominator: Differentiate x⁴eˣ using the product rule, and we get h'(x) = 4x³eˣ + x⁴eˣ

Now, substituting these values into the quotient rule formula, we have:

f'(x) = [(4x³eˣ + x⁴eˣ) * (x⁴eˣ) - (x⁴eˣ) * (4x³eˣ + x⁴eˣ)]/[(x⁴eˣ)^2]

Simplifying further, we get f'(x) = 0