Calculating the Derivative using the Quotient Rule | f(x) = (2x^2 – 1) / (5x + 3)

If f(x)=2x^2−1 / 5x+3, then f′(−1)=

To find the derivative of the function f(x), we can use the quotient rule

To find the derivative of the function f(x), we can use the quotient rule. The quotient rule states that if we have a function of the form f(x) = h(x) / g(x), where h(x) and g(x) are differentiable functions, then the derivative of f(x) is equal to (g(x)*h'(x) – h(x)*g'(x)) / (g(x))^2.

In this case, f(x) = (2x^2 – 1) / (5x + 3). We want to find f'(x) at x = -1, so we will evaluate the derivative of f(x) and substitute x = -1 into the derivative.

Using the quotient rule, the derivative of f(x) is:
f'(x) = [(5x + 3)(4x) – (2x^2 – 1)(5)] / (5x + 3)^2.

Now, we can substitute x = -1 into f'(x) to find f'(-1).
f'(-1) = [(5(-1) + 3)(4(-1)) – (2(-1)^2 – 1)(5)] / (5(-1) + 3)^2
= [(3)(-4) – (2 – 1)(5)] / (3)^2
= [(-12) – (1)(5)] / 9
= [-12 – 5] / 9
= -17 / 9.

Therefore, f'(-1) is equal to -17/9.

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