differentiate y = tan^2(3x) + sec^2(3x)
Use the chain rule separately for each of the two terms. The first term can be differentiated as follows:
Let u = tan 3x and v = u^2
tan^2(3x) = v{u(x)}
d/dx [tan^2(3x)] = (dv/du)*(du/dx)
= 2u * 3 sec^2 3x
= 6 tan(3x)*sec^2(3x)
Use a similar procedure to differentiate sec^2(3x)
To differentiate the function y = tan^2(3x) + sec^2(3x), we will use the chain rule and the derivative formulas for the tangent and secant functions.
Step 1: Apply the chain rule to differentiate tan^2(3x) and sec^2(3x) separately.
The chain rule states that if we have a function h(x) expressed as f(g(x)), the derivative of h(x) with respect to x can be found by multiplying the derivative of f(g(x)) with respect to g(x) and the derivative of g(x) with respect to x.
For tan^2(3x), let's designate f(u) = u^2 and g(x) = tan(3x).
The derivative of f(u) with respect to u is f'(u) = 2u.
The derivative of g(x) with respect to x is g'(x) = 3sec^2(3x).
By applying the chain rule, we can find the derivative of tan^2(3x) as:
(d/dx) tan^2(3x) = 2tan(3x) * 3sec^2(3x) = 6tan(3x)sec^2(3x).
Similarly, for sec^2(3x), let's designate f(u) = u^2 and g(x) = sec(3x).
The derivative of f(u) with respect to u is f'(u) = 2u.
The derivative of g(x) with respect to x is g'(x) = 3sec(3x)tan(3x).
By applying the chain rule, we can find the derivative of sec^2(3x) as:
(d/dx) sec^2(3x) = 2sec(3x) * 3sec(3x)tan(3x) = 6sec^2(3x)tan(3x).
Step 2: Add the derivatives of tan^2(3x) and sec^2(3x) to get the overall derivative.
(d/dx) (tan^2(3x) + sec^2(3x)) = 6tan(3x)sec^2(3x) + 6sec^2(3x)tan(3x).
Therefore, the derivative of y = tan^2(3x) + sec^2(3x) is 6tan(3x)sec^2(3x) + 6sec^2(3x)tan(3x).
To differentiate the function, we utilized the chain rule and the derivative formulas for tangent and secant functions.