$\frac { d } { d x } cos ( x ) - 7 x =$

Explanation:

Step1: Differentiate term by term

Use the sum/difference rule (\frac{d}{dx}(u - v)=\frac{du}{dx}-\frac{dv}{dx}). Let (u = \cos(x)) and (v=7x).

Step2: Differentiate (\cos(x))

The derivative of (\cos(x)) with respect to (x) is (-\sin(x)), i.e., (\frac{d}{dx}\cos(x)=-\sin(x)).

Step3: Differentiate (7x)

Use the power rule (\frac{d}{dx}(ax^n)=anx^{n - 1}). For (y = 7x=7x^1), (a = 7) and (n = 1). Then (\frac{d}{dx}(7x)=7\times1\times x^{1-1}=7).

Step4: Combine the results

(\frac{d}{dx}[\cos(x)-7x]=\frac{d}{dx}\cos(x)-\frac{d}{dx}(7x)=-\sin(x)-7).

Answer:

(-\sin(x)-7)