f(x)=(4x)/(x^(3)+2x^(2)-3x)

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Apply Quotient Rule: To find the derivative of the function f(x)=(4x)/(x^(3)+2x^(2)-3x), we will use the quotient rule. The quotient rule states that if we have a function that is the quotient of two functions, u(x)/v(x), then its derivative is given by (v(x)u'(x) - u(x)v'(x))/(v(x))^2.Identify u(x) and v(x): First, let's identify u(x) and v(x). Here, u(x) = 4x and v(x) = x^3 + 2x^2 - 3x. We will need to find the derivatives of both u(x) and v(x), which are u'(x) and v'(x) respectively.Find u'(x): The derivative of u(x) = 4x with respect to x is u'(x) = 4, since the derivative of x with respect to x is 1.Find v'(x): Now, let's find the derivative of v(x) = x^3 + 2x^2 - 3x. We will use the power rule, which states that the derivative of x^n with respect to x is n*x^(n-1).Apply Power Rule: Applying the power rule to each term of v(x), we get: v'(x) = d/dx(x^3) + d/dx(2x^2) - d/dx(3x)        = 3x^2 + 4x - 3.Calculate v'(x): Now that we have u'(x) and v'(x), we can apply the quotient rule to find the derivative of f(x): f'(x) = (v(x)u'(x) - u(x)v'(x)) / (v(x))^2        = ((x^3 + 2x^2 - 3x)(4) - (4x)(3x^2 + 4x - 3)) / (x^3 + 2x^2 - 3x)^2.Apply Quotient Rule: Let's simplify the numerator of the derivative: Numerator = (4x^3 + 8x^2 - 12x) - (12x^3 + 16x^2 - 12x)            = 4x^3 + 8x^2 - 12x - 12x^3 - 16x^2 + 12x            = -8x^3 - 8x^2.Simplify Numerator: Now we have the simplified numerator and the denominator. The derivative of f(x) is: f'(x) = (-8x^3 - 8x^2) / (x^3 + 2x^2 - 3x)^2.

$f(x)=\frac{4 x}{x^{3}+2 x^{2}-3 x}$

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f(x)=4xx3+2x2−3x f(x)=\frac{4 x}{x^{3}+2 x^{2}-3 x} f(x)=x3+2x2−3x4x​

$f(x)=\frac{4 x}{x^{3}+2 x^{2}-3 x}$