Question

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

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:

\(-\sin(x)-7\)