Find the derivative of f(x) = x^2/2 + cos 5x.

Step-by-step solution

1. 1

   Split the sum. Differentiation is linear, so handle $\dfrac{x^2}{2}$ and $\cos 5x$ independently and add the results.

2. 2

   Differentiate the quadratic term. $\dfrac{d}{dx}\left(\dfrac{x^2}{2}\right)=\dfrac{2x}{2}=x$.

3. 3

   Differentiate the cosine with the chain rule. The inner function is $u=5x$, whose derivative is $5$. Since $\dfrac{d}{du}\cos u=-\sin u$, $\frac{d}{dx}\cos 5x=-\sin(5x)\cdot 5=-5\sin 5x.$ Forgetting the inner factor $5$ — or losing the minus sign — is the usual error here.

4. 4

   Combine the two pieces. $f\'(x)=x-5\sin 5x.$

5. 5

   Verify numerically. At $x=0.7$ the formula gives $0.7-5\sin 3.5=0.7-5(-0.350783)=2.453917$, and a central difference quotient of $f$ at $x=0.7$ agrees to six decimals.