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tan(x-y)=y/(1+x2)
When I take the derivatives of both sides, I get:
sec2(x-y)(1-y')=[(1+x2)y'-2xy]/(1+x2)2
When I take the derivatives of both sides, I get:
sec2(x-y)(1-y')=[(1+x2)y'-2xy]/(1+x2)2
Implicit differentiation is a method used in calculus to find the derivative of a function with respect to a variable that is not explicitly stated in the equation.
Implicit differentiation allows us to find the derivative of functions that cannot be easily solved using traditional methods, such as when the variable is not isolated or when the equation is too complex to be solved algebraically.
The process for implicit differentiation involves treating the dependent variable as a function of the independent variable and using the chain rule and product rule to find the derivative.
Some common mistakes when using implicit differentiation include forgetting to use the chain rule, not properly labeling the variables, and making arithmetic errors.
Implicit differentiation has many real-world applications, such as in physics to find the velocity and acceleration of objects, in economics to find the elasticity of demand, and in engineering to optimize functions and solve optimization problems.