Differentiate
y=[(8x-x^6)/(x^3)]^(-4/5)
My answer:
[4x^3(16+3x^5)]/5 • fifth root of [(8x-x^6)/(x^3)] ^9
y=( (8x-x^6)(x^-3) )^-4/5
y= ( (8x-2 -x^3) )-4/5
y'= -4/5 (8x^-2 -x^-6)^-9/5 * (-16x+6x^5)
I don't think this is the same answer. Check both.
the radicand is a fraction, not a product.
y = [(8x-x^6)/(x^3)]^(-4/5)
y' = (-4/5)[(8x-x^6)/(x^3)]^(-9/5) * -(3x^5+16)/x^3
You can massage that to get the various forms shown here:
http://www.wolframalpha.com/input/?i=derivative+[%288x-x^6%29%2F%28x^3%29]^%28-4%2F5%29
To differentiate the given function y=[(8x-x^6)/(x^3)]^(-4/5), we can use the chain rule along with the power rule and quotient rule. Here's a step-by-step procedure to find the derivative:
Step 1: Simplify the expression inside the square brackets:
[(8x - x^6) / (x^3)] = (8x / x^3) - (x^6 / x^3) = 8 / x^2 - x^3
Step 2: Apply the power rule by multiplying the exponent with each term inside the parentheses and subtracting 1 from the exponent:
(8 / x^2 - x^3)^(-4/5) = 8^(-4/5) / (x^2)^(-4/5) - (x^3)^(-4/5)
= 8^(-4/5) / x^(-8/5) - x^(-12/5)
= 1 / (8^(4/5) * x^(8/5)) - x^(12/5)
Step 3: Differentiate each term using the power rule and the chain rule:
d/dx [1 / (8^(4/5) * x^(8/5))] - d/dx [x^(12/5)]
Step 4: Differentiate the first term:
d/dx [1 / (8^(4/5) * x^(8/5))] = -8/25 * (8^(4/5) * x^(8/5))^(-9/5) * (8/5) * x^(-2/5)
= (-8/25) * (8^(4/5))^(-9/5) * (8/5) * x^(-2/5) * x^(-9/5)
= (-8/25) * (8^(4/5)/5) * x^(-2/5 - 9/5)
= (-8/25) * (8^(4/5)/5) * x^(-11/5)
= -8/25 * (8^(4/5)/5) * (1/x^(11/5))
= -8 * (8^(4/5)/5x^(11/5))
Step 5: Differentiate the second term:
d/dx [x^(12/5)] = (12/5) * x^(12/5 - 1)
= (12/5) * x^(7/5)
= (12/5) * (1/x^(-7/5))
= 12/5x^(-7/5)
Step 6: Combine the results from step 4 and step 5:
dy/dx = -8 * (8^(4/5)/5x^(11/5)) - 12/5x^(-7/5)
= -8 * ((8^(4/5)/5) / x^(11/5)) - 12/5x^(-7/5)
= -8 * (1 / (5x^(11/5))) * (8^(4/5)/5) - 12/5x^(-7/5)
= -8 * (8^(4/5) / (5^2 * x^(11/5))) - 12/5x^(-7/5)
= -8 * (2^4 / (5^2 * x^(11/5))) - 12/5x^(-7/5)
= -2^(4+3) * x^(-7/5) / (5^2 * x^(11/5))
= -16x^(-7/5) / (25x^(11/5))
= -16 / (25 * x^(7/5 + 11/5))
= -16 / (25 * x^(18/5))
Therefore, the derivative of y=[(8x-x^6)/(x^3)]^(-4/5) is -16 / (25 * x^(18/5)).