Differentiate the function
d/d(x) [ (x^{3}-8) • x^{2} + 1/x^{2} - 1 )
rewrite
d/dx [ -1 + 1/x^{2} + x^{2} (-8+x^{3}) ]
differentiate the sum term by term
d/dx (-1) + d/dx (1/x^{2}) + d/dx [x^{2} (-8+x^{3}) ]
derivative of -1 = 0
power rule
d/dx(x^{n}) = n•x^{n-1}
n= -2, d/dx (1/x^{2}) = d/dx (x^{-2}) = -2x^{3}
d/dx = (x^{2} (-8+x^{3})) + (-2/x^{3})
product rule
d/dx (uv)= v du/dx + u dv/dx
u = x^{2}, v = x^{3}-8
= (-2/x^{3}) + (x^{3} - 8) d/dx(x^{2}) +
x^{2 }d/dx(-8+x^{3})
power rule
d/dx(x^{n}) = n•x^{n-1}
n= -2, d/dx (x^{2}) = 2x
d/dx(x^{n}) = n•x^{n-1}
n= -2, d/dx (x^{2}) = 2x
= (-2/x^{3}) + x^{2} (d/dx(-8+x^{3})) +
(-8+x^{3})(2x)
differentiate the sum term by term
= (-2/x^{3}) + (2x)(-8+x^{3}) +
(d/dx(-8) + d/dx(x^{3}) • x^{2}
derivative of -8 is 0
Product rule
= (-2/x^{3}) + (2x)(-8+x^{3}) +
(3x^{2}) • x2
Simplify
= (-2/x^{3}) + (2x)(-8+x^{3}) + 3x^{4}
= (-2/x^{3}) + (-16x) + (2x^{4}) + 3x^{4}
= 5 x^{4} - 2 / x^{3} - 16 x
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