# If #y= sinx/ x^2#, find #dy/dx# and #(d^2y)/(dx^2)#. Then prove that #x^2 (d^2y)/(dx^2) + 4x dy/dx + (x^2 + 2) y = 0# ?

We seek to show that:

And differentiating a second time, we have:

And so, considering the LHS of the given expression:

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To find ( \frac{dy}{dx} ), differentiate ( y = \frac{\sin(x)}{x^2} ) with respect to ( x ) using the quotient rule: [ \frac{dy}{dx} = \frac{(x^2 \cdot \cos(x) - 2x \cdot \sin(x))}{x^4} ]

To find ( \frac{d^2y}{dx^2} ), differentiate ( \frac{dy}{dx} ) with respect to ( x ): [ \frac{d^2y}{dx^2} = \frac{(x^4 \cdot (-\sin(x)) - 4x^3 \cdot \cos(x) - (2x \cdot \cos(x) - 2 \sin(x)))}{x^8} ]

Then, to prove ( x^2 \frac{d^2y}{dx^2} + 4x \frac{dy}{dx} + (x^2 + 2)y = 0 ), substitute ( \frac{dy}{dx} ) and ( \frac{d^2y}{dx^2} ) into the expression and simplify. You should obtain ( 0 ) when the expression is simplified.

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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.

- How do you use the shell method to compute the volume of the solid obtained by rotating the region in the first quadrant enclosed by the graphs of the functions #y=x^2# and #y=2# rotated about the y-axis?
- How do you find all solutions of the differential equation #(d^2y)/(dx^2)=0#?
- How do you find the arc length of the curve #y=x^2/2# over the interval [0, 1]?
- How do you find the volume bounded by #x=2y-y^2# and the line x=0 revolved about the y-axis?
- What is the arc length of #f(x)=(2x^2–ln(1/x+1))# on #x in [1,2]#?

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