How do you use implicit differentiation to find #(d^2y)/(dx^2)# given #4y^2+2=3x^2#?
Implicit differentiation is a special case of the chain rule for derivatives.
Now implicitly differentiating (1) further, we get
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To find (d^2y)/(dx^2) using implicit differentiation, follow these steps:
 Differentiate both sides of the equation with respect to (x) using the chain rule for (y)terms.
 Solve the resulting equation for ((d^2y)/(dx^2)).
Here's the process:
Given: (4y^2 + 2 = 3x^2)

Differentiate both sides with respect to (x): [ \begin{aligned} \frac{d}{dx} (4y^2) + \frac{d}{dx} (2) &= \frac{d}{dx} (3x^2) \ 8yy' &= 6x \ \end{aligned} ]

Now, differentiate (8yy') with respect to (x) again to find ((d^2y)/(dx^2)): [ \begin{aligned} \frac{d}{dx} (8yy') &= \frac{d}{dx} (6x) \ 8(y')^2 + 8y \frac{d^2y}{dx^2} &= 6 \ 8(y')^2 + 8y \frac{d^2y}{dx^2} &= 6 \ \frac{d^2y}{dx^2} &= \frac{6  8(y')^2}{8y} \ \end{aligned} ]
Remember, (y') represents (\frac{dy}{dx}).
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When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
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