Lammy
find the equation of the tangent to the curve y=X^(x^2) - X^(Pi) where x=1 find the equation of the tangent to the curve y=X^(x^2) - X^(Pi) where x=1 @Mathematics
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Annand
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Take log of y and then differentiate...
(dy/dx) at x=1 will give the slope of the tangent...
Lammy
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when you say dy/dx at x=1 do you mean to plug in 1 for all x after i get the derivative of the equation?
Annand
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exactly
Lammy
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is the slop -10.6063
Annand
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you can also find y at x=1...
Thus you know the slope and one point in the line which will help you to find the equation...
Annand
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what was your dy/dx
Annand
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is it not (1-pi)?
Lammy
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yes yes thats the slope right? then how do i find the y to plug in for point slope form
Annand
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you have the equation of the curve... put x=1 in it... that'll give you the point required...
Annand
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that is
y=X^(x^2) - X^(Pi) at x=1
Lammy
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= 0
Annand
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yep
Annand
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so you've got a point and the slope... go on...
Lammy
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can i leave it at y-0=1-pi(x-1)
Annand
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y-0=(1-pi)(x-1)
don't forget the brackets...
Lammy
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thats it?
Annand
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you may write it in the form y=mx+c if you like...
Lammy
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y=-piX +X+pi-1
Annand
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y=(1-pi)x-(1-pi)
Lammy
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thank you soo much!
Annand
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you're welcome...
Annand
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A piece of advice: you should've calculated the derivative by hand....
Lammy
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yea thats the problem i dont know how to do that one
Annand
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do you want me to explain?
Lammy
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yes please
Annand
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In y=X^(x^2) - X^(Pi)
the problem is with the first term... I believe that you know how to differentiate x^(pi)...
Lammy
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yea
Annand
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let y=p-q where
p=X^(x^2) and q=X^(Pi)
Annand
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taking log of p
log(p)=x^2*log(x)
then differentiate
(1/p)dp/dx=x^2/x+2xlog(x)
dp/dx=p[x+2xlog(x)]
dp/dx=x^(x^2)*[x+2xlog(x)]
Annand
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y=dp/dx-dq/dx
Annand
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got it?
Lammy
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(1/p)dp/dx=x^2/x+2xlog(x)
Lammy
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so derivative of log(x) = 2xlog(x)
Annand
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nope...
derivative of log(x) is 1/x
Annand
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p is the product of x^2 and log(x) so you've to use the product rule.
Annand
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\[d(x^2*\log(x))/dx=x^2*d(\log(x))/dx + \log(x)*d(x^2)/dx\]
Lammy
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where is the 1/p from? (1/p)dp/dx=x^2/x+2xlog(x)
Lammy
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where did the x^2 go? dp/dx=p[x+2xlog(x)]
Annand
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x^2/x=x