Final answer:
To find the derivative of the logarithmic function h(x), we use properties of logarithms to express it as a difference and then differentiate using the chain rule, yielding the simplified derivative expression for h'(x).
Step-by-step explanation:
To find the derivative of the function h(x)=log₃(((6x-4)⁶)/((-4x-5)⁸)), we use the properties of logarithms. First, we apply the property that the logarithm of a quotient is the difference of the logarithms of the numerator and the denominator. Next, we use the property that the logarithm of a number raised to an exponent is the product of the exponent and the logarithm of the number.
This leads to:
h(x) = 6*log₃(6x-4) - 8*log₃(-4x-5)
To differentiate, we apply the chain rule:
h'(x) = 6*(1/(6x-4)*ln(3))*(6) - 8*(1/(-4x-5)*ln(3))*(-4)
Simplifying, we get:
h'(x) = 36/(ln(3)*(6x-4)) - (-32)/(ln(3)*(-4x-5))
Note that ln(3) is the natural logarithm of 3 and it is a constant factor.
Finally, we can simplify h'(x) to express the derivative in the simplest form:
h'(x) = 36/(ln(3)*(6x-4)) + 32/(ln(3)*(-4x-5))