Find the 4th derivative of y=4x^3-(2/x). Reads: 4x to the third minus 2 divided by x.
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To find the 4th derivative of the given function y = 4x^3 - (2/x), we'll need to apply the power rule and the quotient rule repeatedly.
Let's start by finding the first derivative of the function:
y' = d/dx(4x^3) - d/dx(2/x)
Using the power rule, we can differentiate 4x^3 term by bringing down the exponent and multiplying by the coefficient:
y' = 12x^2 - d/dx(2/x)
Next, let's differentiate the term (2/x) using the quotient rule:
The quotient rule states that for a function u/v, where u and v are functions of x:
(u/v)' = (u'v - uv') / v^2
In our case, u = 2 and v = x. Thus, we can find the derivative of (2/x) as:
(d/dx)(2/x) = ((d/dx)(2) * x - 2 * (d/dx)(x)) / x^2
Simplifying further:
(d/dx)(2/x) = (0 * x - 2 * 1) / x^2 = -2/x^2
Substituting this back into our previous equation:
y' = 12x^2 - (-2/x^2)
Now, we can simplify further:
y' = 12x^2 + 2/x^2
To find the second derivative (y''), we differentiate y' with respect to x again:
y'' = d/dx(12x^2) + d/dx(2/x^2)
Using the power rule, we find:
y'' = 24x - d/dx(2/x^2)
To find the derivative of 2/x^2, we utilize the quotient rule again:
(d/dx)(2/x^2) = ((d/dx)(2) * x^2 - 2 * (d/dx)(x^2)) / x^4
Expanding further:
(d/dx)(2/x^2) = (0 * x^2 - 2 * 2x) / x^4 = -4x/x^4 = -4/x^3
Substituting into our previous equation:
y'' = 24x - (-4/x^3) = 24x + 4/x^3
Next, we find the third derivative (y''') by taking the derivative of y'' with respect to x:
y''' = d/dx(24x) + d/dx(4/x^3)
Differentiating 24x and -4/x^3:
y''' = 24 - d/dx(4/x^3)
To find the derivative of 4/x^3, we can employ the quotient rule once again:
(d/dx)(4/x^3) = ((d/dx)(4) * x^3 - 4 * (d/dx)(x^3)) / x^6
Expanding further:
(d/dx)(4/x^3) = (0 * x^3 - 4 * 3x^2) / x^6 = -12x^2 / x^6 = -12/x^4
Substituting back into the equation:
y''' = 24 - (-12/x^4) = 24 + 12/x^4
Finally, to find the fourth derivative (y''''), we differentiate y''' with respect to x:
y'''' = d/dx(24) + d/dx(12/x^4)
As 24 is a constant, its derivative is 0. For the second term, we can differentiate 12/x^4 using the power rule:
(d/dx)(12/x^4) = (d/dx)(12) * (x^-4)
Differentiating 12 gives us 0, and the power rule states that d/dx(x^n) = n * x^(n-1). Therefore:
(d/dx)(12/x^4) = 0 * (x^-4) = 0
Since the derivative of 24 is 0 and the derivative of 12/x^4 is 0, we end up with:
y'''' = 0 + 0 = 0
Therefore, the fourth derivative of y = 4x^3 - (2/x) is y'''' = 0.