solve for y' in the function y=sec^3(2x)-3sec2x
Just apply the chain rule
y=sec^3(2x)-3sec2x
y' = 3 sec^2(2x) * sec(2x)tan(2x) * 2 - 3sec(2x)tan(2x)*2
= 6sec(2x)tan(2x) (sec^2(2x)-1)
= 6sec(2x)tan^3(2x)
To solve for y' in the function y = sec^3(2x) - 3sec(2x), we will need to differentiate both sides of the equation with respect to x.
Let's start by finding the derivative of the first term: sec^3(2x).
Using the chain rule, the derivative of sec^3(2x) can be found by multiplying the derivative of sec(2x) with the derivative of the exponent, 3. The derivative of sec(2x) is sec(2x)tan(2x), so we have:
dy/dx = 3sec^2(2x) * sec(2x)tan(2x)
Next, let's find the derivative of the second term: -3sec(2x).
Using the chain rule again, the derivative of -3sec(2x) can be found by multiplying the derivative of sec(2x) with the derivative of the term inside the parentheses, which is 2. Derivative of sec(2x) is sec(2x)tan(2x), so we have:
dy/dx = (-3) * sec(2x)tan(2x) * 2
Combining the two terms, we have:
dy/dx = 3sec^2(2x) * sec(2x)tan(2x) - 6sec(2x)tan(2x)
Simplifying this expression, we get:
dy/dx = sec(2x)(3sec^2(2x) - 6tan(2x))
So, the derivative of y with respect to x, y', is given by:
y' = sec(2x)(3sec^2(2x) - 6tan(2x))
Therefore, the solution for y' in the function y = sec^3(2x) - 3sec(2x) is y' = sec(2x)(3sec^2(2x) - 6tan(2x)).
To solve for y' in the given function y = sec^3(2x) - 3sec(2x), we will use the chain rule of differentiation.
The chain rule states that if we have a composition of functions f(g(x)), the derivative of f with respect to x is given by the derivative of f with respect to g, multiplied by the derivative of g with respect to x.
Let's break down the given function into two functions:
f(x) = sec^3(2x)
g(x) = -3sec(2x)
Now, differentiate each function separately:
f'(x) = 3sec^2(2x) * d(2x)/dx
g'(x) = -3sec(2x) * d(2x)/dx
Note that d(2x)/dx = 2 since the derivative of 2x with respect to x is simply 2 (the derivative of a constant is zero).
Now, we can substitute the calculated derivatives back into the chain rule formula:
y' = f'(g(x)) * g'(x)
= 3sec^2(2x) * (-3sec(2x)) * 2
= -18sec^2(2x)sec(2x)
Therefore, y' = -18sec^2(2x)sec(2x) is the derivative of the given function y = sec^3(2x) - 3sec(2x).