Find the derivative of (3x+17)^13 (14x+6)^11x^-4/5 / (20x +3)^-9
using logarithmic differentiation. Assume x > 0.
y = (3x+17)^13 (14x+6)^11x^-4/5 / (20x +3)^-9
ln y = 13ln(3x+17) + 11 ln(14x+6) - 4/5 ln(x) + 9ln(20x+3)
1/y y' = 39/(3x+7) + 154/(14x+6) - 4/(5x) + 180/(20x+3)
Now the rest is just algebra.
See if you get the result here:
http://www.wolframalpha.com/input/?i=(3x%2B17)%5E13+(14x%2B6)%5E11+x%5E(-4%2F5)+%2F+(20x+%2B3)%5E-9
scroll down some distance to get the derivative.
To find the derivative of the given function using logarithmic differentiation, follow these steps:
Step 1: Take the natural logarithm (ln) of both sides of the equation to simplify the expression and make it more manageable.
ln(y) = ln((3x+17)^13 (14x+6)^11x^(-4/5) / (20x +3)^(-9))
Step 2: Apply the properties of logarithms to expand and simplify the expression.
ln(y) = ln((3x+17)^13) + ln((14x+6)^11x^(-4/5)) - ln((20x +3)^(-9))
Step 3: Use the power rule of logarithms to differentiate each term.
ln(y) = 13 ln(3x+17) + ln((14x+6)^11x^(-4/5)) - 9 ln(20x +3)
Step 4: Apply the logarithmic differentiation rules for the remaining terms.
ln(y) = 13 ln(3x+17) + 11 ln(14x+6) + (-4/5) ln(x) - 9 ln(20x+3)
Step 5: Differentiate both sides of the equation implicitly with respect to x.
(d/dx) ln(y) = (d/dx)(13 ln(3x+17)) + (d/dx)(11 ln(14x+6)) + (d/dx)((-4/5) ln(x)) - (d/dx)(9 ln(20x+3))
Step 6: Apply the chain rule and the derivative of natural logarithm rules to differentiate each term.
(1/y) * (dy/dx) = 13 * (1/(3x+17)) * 3 + 11 * (1/(14x+6)) * 14 + (-4/5) * (1/x) - 9 * (1/(20x+3)) * 20
Step 7: Simplify and solve for (dy/dx) which is the derivative of the given function.
(dy/dx) = y * (13/(3x+17)) * 3 + y * (11/(14x+6)) * 14 + y * (-4/5) * (1/x) - y * (9/(20x+3)) * 20
Step 8: Rewrite y as the original function.
(dy/dx) = (3x+17)^13 (14x+6)^11x^(-4/5) / (20x +3)^(-9) * (13/(3x+17)) * 3 + (3x+17)^13 (14x+6)^11x^(-4/5) / (20x +3)^(-9) * (11/(14x+6)) * 14 + (3x+17)^13 (14x+6)^11x^(-4/5) / (20x +3)^(-9) * (-4/5) * (1/x) - (3x+17)^13 (14x+6)^11x^(-4/5) / (20x +3)^(-9) * (9/(20x+3)) * 20
Simplifying the expression further may require additional algebraic manipulation.