How do you find the derivatives of #y=(3x2)/(4x+3)# by logarithmic differentiation?
see below
Steps in Logarithmic Differentiation:
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To find the derivative of ( y = \frac{3x  2}{4x + 3} ) using logarithmic differentiation, follow these steps:

Take the natural logarithm of both sides of the equation: [ \ln(y) = \ln\left(\frac{3x  2}{4x + 3}\right) ]

Use properties of logarithms to simplify the expression: [ \ln(y) = \ln(3x  2)  \ln(4x + 3) ]

Differentiate both sides of the equation with respect to ( x ): [ \frac{1}{y} \cdot \frac{dy}{dx} = \frac{d}{dx}\left(\ln(3x  2)\right)  \frac{d}{dx}\left(\ln(4x + 3)\right) ]

Use the chain rule to differentiate (\ln(3x  2)) and (\ln(4x + 3)): [ \frac{1}{y} \cdot \frac{dy}{dx} = \frac{1}{3x  2} \cdot \frac{d}{dx}(3x  2)  \frac{1}{4x + 3} \cdot \frac{d}{dx}(4x + 3) ]

Simplify and solve for ( \frac{dy}{dx} ): [ \frac{1}{y} \cdot \frac{dy}{dx} = \frac{3}{3x  2}  \frac{4}{4x + 3} ]

Multiply both sides by ( y ): [ \frac{dy}{dx} = y\left(\frac{3}{3x  2}  \frac{4}{4x + 3}\right) ]

Substitute the original expression for ( y ): [ \frac{dy}{dx} = \frac{3x  2}{4x + 3} \left(\frac{3}{3x  2}  \frac{4}{4x + 3}\right) ]

Simplify the expression to get the final derivative.
[ \frac{dy}{dx} = \frac{(3x  2)(3(4x + 3)  4(3x  2))}{(4x + 3)^2} ]
[ \frac{dy}{dx} = \frac{9(4x + 3)  12x + 8}{(4x + 3)^2} ]
[ \frac{dy}{dx} = \frac{36x + 27  12x + 8}{(4x + 3)^2} ]
[ \frac{dy}{dx} = \frac{24x + 35}{(4x + 3)^2} ]
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When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
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