Question

Find the derivative of the following function and show your work:
f(x)=sin(2x)+cos(3x)−tan(x)

Answer 1

The derivative of f(x) = sin(2x) + cos(3x) - tan(x) can be found by applying the chain and quotient rules to obtain f'(x) = 2cos(2x) - 3sin(3x) - sec^2(x).

Step-by-step explanation:

To find the derivative of the function f(x) = sin(2x) + cos(3x) - tan(x), we will apply the rules of differentiation to each term separately. We use the chain rule for trigonometric functions and the quotient rule for the tangent term:

1. The derivative of sin(2x) is 2cos(2x), because d/dx(sin(u)) = cos(u) * du/dx, where u = 2x.
2. The derivative of cos(3x) is -3sin(3x), because d/dx(cos(u)) = -sin(u) * du/dx, where u = 3x.
3. The derivative of -tan(x) is -sec2(x), because d/dx(tan(u)) = sec2(u) * du/dx, where u = x.

Combining these results, the derivative of the function is:

f'(x) = 2cos(2x) - 3sin(3x) - sec2(x).