The expression `sin^2x - tan x` has to be differentiated.
Using the chain rule, the derivative is:
`2*sin x*cos x - sec^2x`
=> `sin 2x - sec^2x`
The derivative of sin^2x - tan x is sin 2x - sec^2x
Let me try to explain the simplest way to find the derivative of the given function.
we have given the function as sin²x - tanx
what about finding the derivative of the function seperately here.
d(sin²x)/dx - d(tanx)/dx
Now, we know that the derivative of the tanx is sec²x. we only need to find the derivative of sin²x.
We will apply the chain rule here.
as we know chain rule states that inner function derivative * outer function derivative. we have inner function as sin²x and outer function as sinx. i.e.
d(sin²x)/dx = d(sin²x)/du * d(sinx)/dx
we know that we can find the derivative of the variable function only with respect to the same variable.i.e for finding derivative of y^2, we need to derivate it with respect to y. d(y^2)/dy = 2y.
So, since we have d(sin²x)/du. so we need to convert the function sin²x in terms of u to get the derivative.
Let u = sinx then u^2 = sin²x
now, d(sin²x)/du = d(u^2)/du = 2u
We plugged the value u = sinx, so, reverse plugging the value of u = sinx will give.
d(sin²x)/du = d(u^2)/du = 2u = 2sinx
Now, d(sin²x)/dx = d(sin²x)/du * d(sinx)/dx
= 2sinx * d(sinx)/dx
So, the complete derivative of sin²x - tanx will be
d(sin²x)/dx - d(tanx)/dx = 2sinx*cosx - sec²x
Hope this will help you!!