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yKpoI14uk [10]
3 years ago
8

You are given the following equation.

Mathematics
1 answer:
saul85 [17]3 years ago
6 0

Answer:

Step-by-step explanation:

Given the equation  4x²+ 49y² = 196

a) Differentiating implicitly with respect to y, we have;

8x + 98y\frac{dy}{dx} = 0\\98y\frac{dy}{dx}  = -8x\\49y\frac{dy}{dx}  = -4x\\\frac{dy}{dx} = \frac{-4x}{49y}

b)  To solve the equation explicitly for y and differentiate to get dy/dx in terms of x,

First let is make y the subject of the formula from the equation;

If 4x²+ 49y² = 196

49y² = 196 - 4x²

y^{2} =  \frac{196}{49}  - \frac{4x^{2} }{49} \\y = \sqrt{\frac{196}{49}  - \frac{4x^{2} }{49} \\} \\

Differentiating y with respect to x using the chain rule;

Let u=  \frac{196}{49}  - \frac{4x^{2} }{49}

y =  \sqrt{u} \\y =u^{1/2} \\

\frac{dy}{dx}  = \frac{dy}{du} * \frac{du}{dx}

\frac{dy}{du} = \frac{1}{2}u^{-1/2} \\

\frac{du}{dx} =  0 - \frac{8x}{49} \\\frac{du}{dx} =\frac{-8x}{49} \\\frac{dy}{dx} = \frac{1}{2} ( \frac{196}{49}  - \frac{4x^{2} }{49})^{-1/2} *  \frac{-8x}{49}\\\frac{dy}{dx} = \frac{1}{2} (  \frac{196-4x^{2} }{49})^{-1/2} *  \frac{-8x}{49}\\\frac{dy}{dx} = \frac{1}{2} ( \sqrt{ \frac{49}{196-4x^{2} })} *  \frac{-8x}{49}\\\frac{dy}{dx} = \frac{1}{2} *{ \frac{7}\sqrt {196-4x^{2} }} *  \frac{-8x}{49}\\

\frac{dy}{dx} = \frac{-4x}{7\sqrt{196-4x^{2} } }

c) From the solution of the implicit differentiation in (a)

\frac{dy}{dx} = \frac{-4x}{49y}

Substituting y = \sqrt{\frac{196}{49}  - \frac{4x^{2} }{49} \\ into the equation to confirm the answer of (b) can be shown as follows

\frac{dy}{dx} = \frac{-4x}{49\sqrt{\frac{196-4x^{2} }{49} } }\\\frac{dy}{dx}  =  \frac{-4x}{49\sqrt{196-4x^{2}}/7} }\\\\\frac{dy}{dx}  = \frac{-4x}{7\sqrt{196-4x^{2}}}

This shows that the answer in a and b are consistent.

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