Question 231658
I'm assuming the problem is:
Given: {{{y = 4/cos(x)}}}
Find: dy/dx
If this is not correct then then you'll have to repost your question, being clearer about the problem.<br>
If the equation was
{{{y = 4/x = 4x^(-1)}}}
would you know how to find its derivative? (I hope so because the solution to your problem depends on this.) This is a derivative where you bring the exponent down in front (as a coefficient) and then subtract one from the exponent:
{{{dy/dx = (-1)4x^((-1-1)) = -4x^(-2)}}}<br>
Of course we don't actually have {{{y = 4/x}}}, we have {{{y = 4/cos(x)}}}. In a situation like this, where you have a function where it would be nice to just have x, we use the Chain Rule. We find the derivative, treating cos(x) just as if it was x and then we multiply this result by the derivative of cos(x) (which is -sin(x):
{{{y = 4(cos(x))^(-1)}}}
{{{dy/dx = (-1)(4(cos(x))^((-1-1)))*(-sin(x)) = 4sin(x)(cos(x))^(-2) = 4sin(x)/((cos(x))^2)}}}