find the derivative of
1) y=Ln(9x^2-3).e^(7x)
2)y=Ln((6x^3+x^2-x)/(x-1)(x+1))
please help
To find the derivative of each of the given functions, we can use the rules of differentiation. The two functions involve logarithmic and exponential functions, as well as polynomial functions. We can differentiate them step by step using these rules.
1) y = Ln(9x^2 - 3)e^(7x)
Step 1: Differentiate the logarithmic function Ln(9x^2 - 3) and leave the exponential function e^(7x) as it is. The derivative of a natural logarithm function is given by 1 divided by the argument, multiplied by the derivative of the argument.
So, the derivative of Ln(9x^2 - 3) is (1 / (9x^2 - 3)) * (d/dx)(9x^2 - 3).
Step 2: Differentiate the exponential function e^(7x) using the chain rule. The derivative of e^(7x) is e^(7x) times the derivative of the exponent 7x, which is 7.
So, the derivative of e^(7x) is 7e^(7x).
Step 3: Multiply the derivative of the logarithmic function from Step 1 with the exponential function from Step 2.
Therefore, the derivative of y = Ln(9x^2 - 3)e^(7x) is (1 / (9x^2 - 3)) * (d/dx)(9x^2 - 3) * 7e^(7x).
2) y = Ln((6x^3 + x^2 - x)/(x-1)(x+1))
Step 1: Simplify the given function inside the natural logarithm. The expression can be rewritten as:
y = Ln( [x(6x^2 + x - 1)] / (x^2 - 1) ).
Step 2: Now, differentiate the simplified expression. The derivative of a natural logarithm function is given by 1 divided by the argument, multiplied by the derivative of the argument.
So, the derivative of Ln( [x(6x^2 + x - 1)] / (x^2 - 1) ) is (1 / [x(6x^2 + x - 1) / (x^2 - 1)]) * (d/dx)[x(6x^2 + x - 1) / (x^2 - 1)].
Step 3: Apply the quotient rule to differentiate the expression [x(6x^2 + x - 1) / (x^2 - 1)].
The quotient rule states that the derivative of f/g is given by [g(df/dx) - f(dg/dx)] / g^2, where f and g are functions of x.
So, for our expression [x(6x^2 + x - 1) / (x^2 - 1)], let f = x(6x^2 + x - 1) and g = (x^2 - 1).
Differentiating f, we have df/dx = (d/dx)(x(6x^2 + x - 1)), and for g, we have dg/dx = (d/dx)(x^2 - 1).
Now, plug these values into the quotient rule formula and simplify the expression.
Therefore, the derivative of y = Ln((6x^3 + x^2 - x)/(x-1)(x+1)) can be found using the quotient rule.