The value of dz/dt using the chain rule is [tex]2t[(t^2-1)^9 -2(t^4-1)]+2t[9(t^2+1)(t^2-1)^8-(t^2+1)^2][/tex]
According to chain rule:
[tex]\frac{dz}{dt} = \frac{\delta z}{\delta x} \frac{dx}{dt} + \frac{\delta z}{\delta y} \frac{dy}{dt}[/tex]
Given the following expressions
z = xy⁹ − x²y,
x = t² + 1
y = t² − 1
Get the differentials in the equation
[tex]\delta z/\delta x = y^9-2xy\\dzx/dt = 2t\\\delta z/\delta y =9xy^8-x^2\\dy/dt = 2t[/tex]
Substitute the differentials into the formula
[tex]\frac{dz}{dt} = (y^9-2xy)(2t)+(9xy^8-x^2)(2t)\\\frac{dz}{dt} = 2t[(t^2-1)^9 -2(t^4-1)]+2t[9(t^2+1)(t^2-1)^8-(t^2+1)^2][/tex]
Hence the value of dz/dt using the chain rule is [tex]2t[(t^2-1)^9 -2(t^4-1)]+2t[9(t^2+1)(t^2-1)^8-(t^2+1)^2][/tex]
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