Differentiate with respect to x. y=7e^3x/2x-1. write in simplified factored form.
To differentiate the function y = 7e^(3x) / (2x - 1) with respect to x, you can use the quotient rule.
The quotient rule states that for a function of the form f(x) = u(x) / v(x), the derivative of f(x) with respect to x is given by:
f'(x) = [u'(x) * v(x) - u(x) * v'(x)] / v(x)^2
To apply the quotient rule to the given function, we need to find the derivatives of the numerator and denominator separately.
Numerator (u(x)):
u(x) = 7e^(3x)
To differentiate this, we use the chain rule:
u'(x) = 7 * d/dx[e^(3x)]
= 7 * 3e^(3x)
= 21e^(3x)
Denominator (v(x)):
v(x) = 2x - 1
The derivative of v(x) with respect to x is simply the coefficient of x, which is 2.
Now we can plug these values into the quotient rule formula to find the derivative of y.
y' = [u'(x) * v(x) - u(x) * v'(x)] / v(x)^2
= [21e^(3x) * (2x - 1) - 7e^(3x) * 2] / (2x - 1)^2
Now, to simplify the factored form of the expression, we can distribute and collect like terms.
y' = [42xe^(3x) - 21e^(3x) - 14e^(3x)] / (2x - 1)^2
= [42xe^(3x) - 35e^(3x)] / (2x - 1)^2
= 7e^(3x) * (6x - 5) / (2x - 1)^2
Therefore, the simplified factored form of the derivative of y is 7e^(3x) * (6x - 5) / (2x - 1)^2.