Question

Find the derivative of the function g(x) = e^x / x⁶.

Answer 1

The derivative of g(x) = e^x / x^6 is found using the quotient rule. After applying the quotient rule, the derivative simplifies to e^x * (x - 6) / x^6.

Step-by-step explanation:

To find the derivative of the function g(x) = e^x / x^6, we will use the quotient rule. The quotient rule is a method for finding the derivative of a function that is the quotient of two other functions. In this case, the numerator is e^x (where e is the base of the natural logarithm), and the denominator is x^6.

The quotient rule states that if we have a function h(x) = f(x)/g(x), then its derivative h'(x) is given by:

h'(x) = (g(x) * f'(x) - f(x) * g'(x)) / g(x)^2

Applying this to our function:

1. Let f(x) = e^x, which has the derivative f'(x) = e^x (since the derivative of e^x with respect to x is e^x).
2. Let g(x) = x^6, which has the derivative g'(x) = 6x^5 (using the power rule).
3. Apply the quotient rule:

g'(x) = (x^6 * e^x - e^x * 6x^5) / x^12

Simplify:

g'(x) = (e^x * x^6 - 6e^x * x^5) / x^12

g'(x) = e^x * (x - 6) / x^6

So the derivative of g(x) is e^x * (x - 6) / x^6.