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the_ace
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1. use implicit differentiation to evaluate y(prime) for x^3+y^3=4xy+1 at the point (2,1)
Implicit differentiation is a method used to find the derivative of a function that is not explicitly written in terms of one variable. It involves treating one variable as the dependent variable and differentiating with respect to the other variable, while treating the dependent variable as a function of the independent variable.
To use implicit differentiation, we treat y as a function of x and use the rules of differentiation to solve for dy/dx. For this particular equation, we would first take the derivative of both sides with respect to x, then apply the chain rule to the y terms, and finally solve for dy/dx.
After applying implicit differentiation, we will have an expression for dy/dx in terms of x and y. This expression can then be used to find the slope of the curve at any given point on the original equation.
Yes, implicit differentiation can be applied to any equation that is not explicitly written in terms of one variable. However, the resulting derivative may not always be easy to solve or interpret.
Implicit differentiation is commonly used in physics and engineering to solve problems involving related rates, optimization, and curves with changing slopes. It is also used in economics and finance to analyze supply and demand curves and marginal revenue curves.