(a) Find y^' by implicit differentiation. (b) Solve the...

(a) Find $y^{\prime}$ by implicit differentiation. (b) Solve the equation explicitly for $y$ and differentiate to get $y^{\prime}$ in terms of $x$. (c) Check that your solutions to parts (a) and (b) are consistent by substituting the expression for $y$ into your solution for part (a). $5 x^{2}-y^{3}=7$

(a) Find $y^{\prime}$ by implicit differentiation.
(b) Solve the equation explicitly for $y$ and differentiate to get $y^{\prime}$ in terms of $x$.
(c) Check that your solutions to parts (a) and (b) are consistent by substituting the expression for $y$ into your solution for part (a).
$5 x^{2}-y^{3}=7$

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Key Concepts

Implicit Differentiation

A technique used when dealing with equations where y is defined implicitly as a function of x. Instead of isolating y, both sides of the equation are differentiated with respect to x, applying the chain rule to account for y as a function of x.

Explicit Differentiation

After solving an equation for y in terms of x (i.e., expressing y explicitly), differentiation can be done directly using standard rules, yielding the derivative of y with respect to x in terms of x.

Chain Rule

A fundamental differentiation rule used when differentiating composite functions. In the context of implicit differentiation, it is used to differentiate terms that involve y, by multiplying the derivative of the outer function with the derivative of y with respect to x.

Algebraic Manipulation

The method of rearranging and solving equations to isolate a variable, which is essential when converting an implicit equation into an explicit one for further analysis.

Consistency Verification

A process to ensure that independently obtained results from different differentiation methods are equivalent. This is done by substituting the explicit expression for y into the derivative derived by implicit differentiation, confirming the validity of both methods.

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