If $p(x)=3x\cos(x)+2x$, find $p'(0)$

Expert intro: This problem uses the product rule and then evaluates the derivative at a specific point. The main goal is to differentiate correctly before substituting $x=0$.

Detailed walkthrough

Step 1: Differentiate $p(x)$

Given

$p(x)=3x\cos(x)+2x$

use the product rule on $3x\cos(x)$:

$\frac{d}{dx}[3x\cos(x)] = 3\cos(x)-3x\sin(x)$

and

$\frac{d}{dx}[2x]=2$

So

$p'(x)=3\cos(x)-3x\sin(x)+2$

Step 2: Evaluate at $x=0$

$p'(0)=3\cos(0)-3(0)\sin(0)+2$

Since $\cos(0)=1$ and $\sin(0)=0$,

$p'(0)=3(1)-0+2=5$

Final answer

$5$

💡 Pitfall guide

A frequent error is forgetting the negative sign that appears when differentiating $\cos(x)$ to get $-\sin(x)$. Another common slip is evaluating before differentiating the full expression.

🔄 Real-world variant

If the function were $p(x)=3x\sin(x)+2x$, the product rule would still apply, but the trig derivative would change and the final value at $x=0$ would be different.

🔍 Related terms

product rule, trigonometric derivative, instantaneous rate of change