(3/x)^2 how to find the derative of this fraction. having trouble with getting the answers. i need a walkthrough...
power rule
9x^(-2)
-18x^(-3)
To find the derivative of the fraction (3/x)^2, you can use the quotient rule. Let's break down the process step by step:
Step 1: Write down the fraction equation.
f(x) = (3/x)^2
Step 2: Apply the power rule to simplify the expression.
f(x) = (3^2) / (x^2)
= 9 / x^2
Step 3: Now, we can differentiate using the quotient rule. The quotient rule states that if you have a fraction u(x)/v(x), the derivative is calculated as:
(v(x) * u'(x) - u(x) * v'(x)) / (v(x))^2
Step 4: Identify u(x), v(x), u'(x), and v'(x) in the equation f(x) = 9 / x^2.
u(x) = 9
v(x) = x^2
u'(x) = 0 (since a constant's derivative is always zero)
v'(x) = 2x
Step 5: Substitute the values into the quotient rule formula.
f'(x) = (x^2 * 0 - 9 * 2x) / (x^2)^2
= (-18x) / x^4
= -18 / x^3
Therefore, the derivative of the fraction (3/x)^2 is -18 / x^3.