Math Is Fun Forum

fcb1012

Member

Registered: 2007-01-24

Posts: 2

cal 2 help

ok i have 1 medium problem and 2 difficult problems i cant seem to figure out. for all 3 i am wanting to find the derivative.

1) cosx^2=xe^y

2) y=x(arcsinx)^2 - 2x + 2sqrt(1-x^2)arcsin(x)

3) y= sqrt(x^2-4) - 2arcsec(x/2) , 2<x<4

mathsyperson

Moderator

Registered: 2005-06-22

Posts: 4,900

Re: cal 2 help

The first problem needs a combination of implicit differentiation and the product rule.

The left side can be differentiated fairly easily, and it becomes -2x sinx². You can either do that in one step or use the chain rule, but they'll both give you that answer.

The right side is a bit trickier. First we split it into its two products: x * e^y.

Now we can use the product rule to differentiate it. This becomes e^y + xe^y dy/dx.
This is because to differentiate e^y with respect to x, you need to differentiate it with respect to y and stick a dy/dx on the end.

So now we have -2x sinx² = e^y + xe^y dy/dx.

We can manipulate this to get that dy/dx = -(2x sinx² + e^y)/xe^y.
To get it in terms of x, we can substitute xe^y = cosx², which gives us dy/dx = -(2x tanx² + 1/x)

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Why did the vector cross the road?
It wanted to be normal.