Finding Derivatives Implicitly and Explicitly In Exercises...

Finding Derivatives Implicitly and Explicitly In Exercises $21-24,$ (a) find two explicit functions by solving the equation for $y$ in terms of $x,($ b) sketch the graph of the equation and label the parts differentiate the explicit functions, and (d) find $d y / d x$ implicitly and show that the result is equivalent to that of part (c). $$ x^{2}+y^{2}=64 $$

Key Concepts

Graphing of Implicit Curves

Graphing involves visualizing the curve defined by an implicit equation. By interpreting the explicit forms and understanding the overall shape of the curve, one can divide the graph into distinguishable parts, label sections corresponding to different branches, and better understand the geometry of the solution.

Implicit Differentiation

This concept involves differentiating an equation that defines a relationship between variables without first solving for one variable explicitly in terms of the other. By applying the chain rule when differentiating terms that include the dependent variable, the derivative is found in an implicit manner, allowing one to handle curves that are not easily expressed as explicit functions.

Explicit Differentiation

This refers to finding the derivative of a function when it is presented in the form y = f(x). Once the equation is solved to express y explicitly, standard differentiation techniques, such as power rule and chain rule, can be applied directly to compute the derivative.

Conversion from Implicit to Explicit Form

This key concept involves solving the implicit equation algebraically to obtain one or more explicit functions. It allows one to separate the different branches (or segments) of the graph and provides a means to apply explicit differentiation directly to each branch.

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