High School Calculus/Implicit Differentiation

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Implicit Differentiation[edit]

When a functional relation between x and y cannot be readily solved for y, the preceding rules may be applied directly to the implicit function.

The derivative will usually contain both x and y. Thus the derivative of an algebraic function, defined by setting the polynomial of x and y to zero.

Ex. 1

Given the function y of x


Find {\operatorname{d}y\over\operatorname{d}x}




In solving for {\operatorname{d}y\over\operatorname{d}x} we must first factor the differentiation problem

In doing this we get


From here we subtract the {\operatorname{d}y\over\operatorname{d}x} to one side

Thus giving us


Here I am going to skip a step in solving this implicit differentiation problem. I am going to skip the step where I divide the -1 over to the other side.

From here we divide the polynomial from the \operatorname{d}y\over\operatorname{d}x over to the other side. Giving us

\left (\frac{-5x^4+5y}{-5x+5y^4}\right)={\operatorname{d}y\over\operatorname{d}x}

Now we simplify and get

{\operatorname{d}y\over\operatorname{d}x}=\left (\frac{x^4-y}{x-y^4}\right)

Other problems to work on

Ex. 2

Find {\operatorname{d}y\over\operatorname{d}x} given the function


Ex. 3

Find {\operatorname{d}y\over\operatorname{d}x} given the function