find the derivative of the function
f(x) = e^(3x) sec (7x)
To find the derivative of the function f(x) = e^(3x) sec(7x), we can use the product rule.
Step 1: Determine the derivative of e^(3x).
The derivative of e^(kx) with respect to x is ke^(kx), where k is a constant. In this case, k = 3, so the derivative of e^(3x) is 3e^(3x).
Step 2: Determine the derivative of sec(7x).
The derivative of sec(x) with respect to x is sec(x)tan(x), so the derivative of sec(7x) with respect to x is sec(7x)tan(7x).
Step 3: Apply the product rule.
To find the derivative of the product of two functions, we apply the product rule. The product rule states that if we have two functions u(x) and v(x), the derivative of their product uv with respect to x is given by uv' + vu'.
Applying the product rule, we get:
f'(x) = (3e^(3x))(sec(7x)tan(7x)) + (e^(3x))(sec(7x)tan(7x))
Simplifying this expression further, we can factor out (e^(3x))(sec(7x)tan(7x)):
f'(x) = (e^(3x))(sec(7x)tan(7x))(3 + 1)
Simplifying the coefficient further, we get:
f'(x) = 4(e^(3x))(sec(7x)tan(7x)), which is the derivative of the function f(x) = e^(3x) sec(7x).