Question

question
find f(x) where f(x) = 1/3x + 3cos(x).
provide your answer below:
f(x) = □

Explanation:

Step1: Apply sum - rule of differentiation

The sum - rule states that if $f(x)=u(x)+v(x)$, then $f^{\prime}(x)=u^{\prime}(x)+v^{\prime}(x)$. Here $u(x)=\frac{1}{3}x$ and $v(x)=3\cos(x)$.

Step2: Differentiate $u(x)=\frac{1}{3}x$

Using the power - rule $\frac{d}{dx}(ax)=a$ (where $a = \frac{1}{3}$), we get $u^{\prime}(x)=\frac{1}{3}$.

Step3: Differentiate $v(x)=3\cos(x)$

Using the constant - multiple rule $\frac{d}{dx}(cf(x))=c\frac{d}{dx}(f(x))$ and $\frac{d}{dx}(\cos(x))=-\sin(x)$, we have $v^{\prime}(x)=3\times(-\sin(x))=- 3\sin(x)$.

Step4: Combine the derivatives

By the sum - rule, $f^{\prime}(x)=u^{\prime}(x)+v^{\prime}(x)=\frac{1}{3}-3\sin(x)$.

Answer:

$\frac{1}{3}-3\sin(x)$