Differentiate the function.((x^2)-((3x^2)(ln(4x))))/(x^6)
let's skip some of those brackets for easier reading,
((x^2)-((3x^2)(ln(4x))))/(x^6)
= [ x^2 - 3x^2 ln(4x) ]/ x^6
= [ 1 - 3 ln(4x) ] / x^4
= x^-4 - (3x^-4)(ln (4x))
dy/dx = -4x^-5 - ( (3x^-4)(1/x) - (12x^-5)ln(4x) )
= -4x^-5 - 3x^-5 + 12x^-5(ln(4x))
= x^-5( -4 -3 + 12ln(4x))
= -(7 - 12ln(4x))/x^5
or
= (12ln(4x) - 7)/x^5
confirmation:
http://www.wolframalpha.com/input/?i=dy%2Fdx+for++y%3D+((x%5E2)-((3x%5E2)(ln(4x))))%2F(x%5E6)
https://www.derivative-calculator.net
To differentiate the function f(x) = ((x^2) - ((3x^2)(ln(4x)))) / (x^6), we can use the quotient rule of differentiation. The quotient rule states that if we have a function in the form f(x) = g(x) / h(x), then its derivative is given by:
f'(x) = (g'(x) * h(x) - g(x) * h'(x)) / (h(x))^2
Let's differentiate the function step by step using the quotient rule.
Step 1: Identify the functions g(x) and h(x).
g(x) = (x^2) - ((3x^2)(ln(4x)))
h(x) = x^6
Step 2: Find the derivatives of g(x) and h(x).
We need to find g'(x) and h'(x) using the power rule and the product rule.
g'(x) = 2x - (6x)(ln(4x)) - 3x^2 * (1/x) * (1/(4x)) [Applying the product rule and chain rule]
= 2x - 6x ln(4x) - 3 * (1/x) * (1/(4x^2))
= 2x - 6x ln(4x) - 3 / (4x^3)
h'(x) = 6x^5 [Applying the power rule]
Step 3: Substitute the values into the quotient rule formula.
f'(x) = (g'(x) * h(x) - g(x) * h'(x)) / (h(x))^2
= ((2x - 6x ln(4x) - 3 / (4x^3)) * (x^6) - ((x^2) - (3x^2)(ln(4x))) * (6x^5)) / (x^6)^2
= (2x^7 - 6x^7 ln(4x) - 3x^3 + 12x^8 ln(4x) - 6x^8 ln(4x))/(x^12)
Simplifying further, we get:
f'(x) = (2x^7 + 12x^8 ln(4x) - 6x^7 ln(4x) - 3x^3) / (x^12)
Therefore, the derivative of the given function is f'(x) = (2x^7 + 12x^8 ln(4x) - 6x^7 ln(4x) - 3x^3) / (x^12).