Answer:
d⁴y/dx⁴ = -12/x⁴
Step-by-step explanation:
y = 2In x
We want to find d⁴y/dx⁴. This is the fourth derivative.
Using product rule, first derivative is;
dy/dx = 2/x
Using quotient rule, we can find the second, third and fourth derivative;
d²y/dx² = -2/x²
d³y/dx³ = 4/x³
d⁴y/dx⁴ = -12/x⁴
Given that we know the relation y = f(x), we want to get the fourth differentiation.
We will get:
[tex]\frac{d^4y}{dx^4} = \frac{-12}{x^4}[/tex]
So we start by knowing that:
[tex]y = 2*ln(x)[/tex]
Then the differentiations are:
[tex]\frac{dy}{dx} = 2*\frac{1}{x} = \frac{2}{x} \\\\\frac{d^2y}{dx^2} = -2*\frac{1}{x^2} = \frac{-2}{x^2} \\\\\frac{d^3y}{dx^3} = (-2)*\frac{-2}{x^3} = \frac{4}{x^3} \\\\\frac{d^4y}{dx^4} = (-3)*\frac{4}{x^4} = \frac{-12}{x^4}[/tex]
So the fourth differentiation of y is just:
[tex]\frac{d^4y}{dx^4} = \frac{-12}{x^4}[/tex]
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