Question

find the derivative of ( f(x) ).
( f(x)=7 sin (x)-5 cos (x) )
( f^{prime}(x)= )

Explanation:

Step1: Recall derivative rules

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

Step2: Differentiate term by term

For the first term \(7\sin(x)\), using the constant multiple rule and the derivative of \(\sin(x)\), we get \(7\frac{d}{dx}[\sin(x)] = 7\cos(x)\).
For the second term \(-5\cos(x)\), using the constant multiple rule and the derivative of \(\cos(x)\), we get \(-5\frac{d}{dx}[\cos(x)]=-5(-\sin(x)) = 5\sin(x)\).

Step3: Combine the derivatives

Adding the derivatives of the two terms together, we have \(f'(x)=7\cos(x)+5\sin(x)\).

Answer:

\(7\cos(x) + 5\sin(x)\)