Find constants to make a derivative equation true

Let

$f(x) = (ax+b)\sin x + (cx + d) \cos x.$

Find values for the constants $a,b,c,d$ such that

$f'(x) = x \cos x.$

We evaluate the derivative,

$\begin{align*} f'(x) &= a \sin x + (ax+b)\cos x + c \cos x + (cx+d)(-\sin x)\\ &= (a-d-cx)\sin x + (ax+b+c) \cos x. \end{align*}$

Then, settings this equal to $x \cos x$ we must have

$a-d-cx = 0 \qquad \text{and} \qquad ax+b+c = x.$

Equating like powers of $x$ in the equation on the right, we have $a = 1$ and $b+c = 0$ . In the equation on the left we have $c = 0$ and $a-d = 0$ . But $a-d = 0$ implies $d = 1$ since we already know $a = 1$ . Finally, since $b+c = 0$ and $c = 0$ we must have $b = 0$ as well. Putting this all together we have,