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hram777 [196]
3 years ago
8

What is the value of \dfrac{d}{dx}\left(\dfrac{2x+3}{3x^2-4}\right) dx d ​ ( 3x 2 −4 2x+3 ​ )start fraction, d, divided by, d, x

, end fraction, (, start fraction, 2, x, plus, 3, divided by, 3, x, squared, minus, 4, end fraction, )at x=-1x=−1x, equals, minus, 1 ?
Mathematics
1 answer:
stellarik [79]3 years ago
6 0

Answer:

4.

Step-by-step explanation:

We are asked to find the value of expression \frac{d}{dx}(\frac{2x+3}{3x^2-4}) at x=-1.

First of all, we will find the derivative of the given expression using "Quotient Rule of Derivatives" as shown below:

(\frac{f(x)}{g(x)})'=\frac{f'(x)\cdot g(x)-f(x)\cdot g'(x)}{(g(x))^2}

\frac{d}{dx}(\frac{2x+3}{3x^2-4})

\frac{\frac{d}{dx}(2x+3)*(3x^2-4)-(2x+3)*\frac{d}{dx}(3x^2-4)}{(3x^2-4)^2}

\frac{2*(3x^2-4)-(2x+3)*(6x)}{(3x^2-4)^2}

\frac{6x^2-8-12x^2-18x}{(3x^2-4)^2}

\frac{-6x^2-18x-8}{(3x^2-4)^2}

Therefore, our required derivative is \frac{-6x^2-18x-8}{(3x^2-4)^2}.

Now, we will substitute x=-1 in our derivative to find the required value as:

\frac{-6(-1)^2-18(-1)-8}{(3(-1)^2-4)^2}

\frac{-6(1)+18-8}{(3(1)-4)^2}

\frac{-6+18-8}{(3-4)^2}

\frac{4}{(-1)^2}

\frac{4}{1}

4

Therefore, the value of expression \frac{d}{dx}(\frac{2x+3}{3x^2-4}) at x=-1 is 4.

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