Find dy/dx by implicit differentiation given that
tan(4x+y)=4x
sec^2(4x+y) * (4+y') = 4
4+y' = 4cos^2(4x+4y)
y' = 4cos^2(4x+4y)-4 = -4sin^2(4x+4y)
sec^2(4x+y) * (4+y') = 4
4+y' = 4cos^2(4x+4)
y' = 4cos^2(4x+y)-4 = -4sin^2(4x+y)
Then sin^2(4x+y)=(4x/sqrt(16x^2+1))^2
=16x^2/(16x^2+1)
So y' = -4(16x^2/(16x^2+1))
=-64x^2/(16x^2+1))
from the ans choices i choose:
dy/dx= - 4x^2/x^2+1
is that right
Looks good to me.
To find dy/dx by implicit differentiation, we will differentiate both sides of the equation with respect to x, treating y as a function of x.
Let's start by differentiating the left side of the equation using the chain rule. The derivative of tan(u), where u is a function of x, can be given as sec^2(u) * du/dx. In this case, u = 4x + y, so du/dx = d(4x+y)/dx.
Differentiating the right side of the equation is straightforward since it is simply the derivative of 4x with respect to x, which is 4.
Using the chain rule on the left side:
sec^2(4x+y) * d(4x+y)/dx = 4.
Now, let's solve for d(4x+y)/dx, which is what we're looking for:
d(4x+y)/dx = 4 / sec^2(4x+y).
Simplifying further, sec^2 is equal to 1/cos^2, so we can write the equation as:
d(4x+y)/dx = 4 * cos^2(4x+y).
Therefore, dy/dx, the derivative of y with respect to x, is:
dy/dx = 4 * cos^2(4x+y).
And that is the answer.