Question

find the derivative of ( f(x) ).

( f(x)=-2 cos (x)+10 sin (x) )

( f^{prime}(x)= )

Explanation:

Step1: Recall derivative rules

The derivative of \(\cos(x)\) is \(-\sin(x)\), and the derivative of \(\sin(x)\) is \(\cos(x)\). Also, the derivative of a constant multiple of a function is the constant multiple of the derivative.

Step2: Differentiate \(-2\cos(x)\)

Using the constant multiple rule and the derivative of \(\cos(x)\), the derivative of \(-2\cos(x)\) is \(-2\times(-\sin(x)) = 2\sin(x)\).

Step3: Differentiate \(10\sin(x)\)

Using the constant multiple rule and the derivative of \(\sin(x)\), the derivative of \(10\sin(x)\) is \(10\times\cos(x)=10\cos(x)\).

Step4: Combine the derivatives

Adding the derivatives from Step2 and Step3, we get \(f'(x)=2\sin(x)+10\cos(x)\).

Answer:

\(2\sin(x) + 10\cos(x)\)