dy/dx[tan(y)=x=y] in terms of y
the answer my teacher got was cot^2(y) but i get is 1/[sec^2(y)-1]
either i made a mistake, my teacher made a mistake or we're both right and i just need to simplify.
please tell me which of the following is the case and how to get to the correct answer.
let's see if cot^2 y = 1/(sec^2 y - 1 )
RS = 1/(1/cos^2y - 1)
= 1/((1 - cos^2y)/cos^2y)
= 1/((sin^2y/cos^2))
= cos^2y/sin^2y
= cot^2 y
so you are both right!
ok, thank you. is it preferable to change it like that? cause i honestly have no idea why it would matter.
oh and just to be clear you got this using trig identities right?
To find dy/dx, we can use implicit differentiation. Let's start with the given equation:
tan(y) = x = y
To differentiate both sides of the equation with respect to x, we will apply the chain rule. Differentiating x with respect to x gives us 1, and differentiating y with respect to x gives us dy/dx. Remember that when differentiating tan(y), we use the chain rule and derivative of tan(u) is sec^2(u) * du/dx. Applying these rules, we can differentiate both sides of the equation:
sec^2(y) * dy/dx = 1
To solve for dy/dx, we isolate the term by dividing both sides of the equation by sec^2(y):
dy/dx = 1 / sec^2(y)
Recall that sec^2(y) is equal to 1 + tan^2(y). Substituting this value, we get:
dy/dx = 1 / (1 + tan^2(y))
Now, let's simplify this expression. Using the trigonometric identity 1 + tan^2(y) = sec^2(y), we can rewrite the expression:
dy/dx = 1 / sec^2(y) = cos^2(y)
Finally, using the identity cos^2(y) = 1 / (1 + tan^2(y)), we can simplify our expression further:
dy/dx = 1 / (1 + tan^2(y)) = 1 / (sec^2(y) + tan^2(y))
Therefore, the correct answer is dy/dx = 1 / (sec^2(y) + tan^2(y)), or you could also rewrite it as dy/dx = 1 / (1 + cot^2(y)). Both are equivalent expressions. It appears that your teacher made a mistake, and the correct answer is indeed 1 / (sec^2(y) + tan^2(y)).