Find the derivative by solving the given implicit equation for y explicitly in terms of x.

Question

Find the derivative  by solving the given implicit equation for y explicitly in terms of x. ​  ​

Quizplus · Accepted Answer

To find the derivative, we need to first solve the given implicit equation for y explicitly in terms of x.

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We can rewrite this equation as:

y^2 + x^2 = 25

Solving for y, we get:

y = ±sqrt(25 - x^2)

Taking the derivative of y with respect to x, we get:

dy/dx = ±(-x/sqrt(25 - x^2))

Simplifying this expression, we get:

dy/dx = ±(-x/|y|)

Since we have both positive and negative values for y, we have to consider both cases. However, we can see that the denominator |y| is always positive, so we can drop the absolute value sign.

Therefore, the derivative is:

dy/dx = -x/y

Plugging in our expression for y, we get:

dy/dx = -x/±sqrt(25 - x^2)

Simplifying the sign, we get:

dy/dx = -x/sqrt(25 - x^2)

This matches choice (C).

Find the Derivative by Solving the Given Implicit Equation