How do you use implicit differentiation to find #(d^2y)/(dx^2)# given #4y^2+2=3x^2#?

Answer 1

#(d^2y)/(dx^2)=3/(8y)[2-(3x^2)/(2y^2)]#

Implicit differentiation is a special case of the chain rule for derivatives.

Generally differentiation problems involve functions i.e. #y=f(x)# - written explicitly as functions of #x#. However, some functions y are written implicitly as functions of #x#.
So what we do is to treat #y# as #y=y(x)# and use chain rule. This means differentiating #y# w.r.t. #y#, but as we have to derive w.r.t. #x#, as per chain rule, we multiply it by #(dy)/(dx)#.
Hence implicit differentiating #4y^2+2=3x^2#, we get
#4xx2yxx(dy)/(dx)+0=3xx2x#
or #8y(dy)/(dx)=6x# ........................(1)
and #(dy)/(dx)=(6x)/(8y)=(3x)/(4y)# ........................(2)

Now implicitly differentiating (1) further, we get

#8(dy)/(dx)xx(dy)/(dx)+8yxx(d^2y)/(dx^2)=6#
or using (2) we get #8((3x)/(4y))^2+8y(d^2y)/(dx^2)=6#
or #8(9x^2)/(16y^2)+8y(d^2y)/(dx^2)=6#
or #(d^2y)/(dx^2)=1/(8y)[6-(9x^2)/(2y^2)]=3/(8y)[2-(3x^2)/(2y^2)]#
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Answer 2

To find (d^2y)/(dx^2) using implicit differentiation, follow these steps:

  1. Differentiate both sides of the equation with respect to (x) using the chain rule for (y)-terms.
  2. Solve the resulting equation for ((d^2y)/(dx^2)).

Here's the process:

Given: (4y^2 + 2 = 3x^2)

  1. 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} ]

  2. 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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Answer from HIX Tutor

When evaluating a one-sided 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 one-sided 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 one-sided 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 one-sided 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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