Q.Find the derivative of $y=x\sin x+\cos x$.

Combining the algebra of

derivatives with the derivatives of the two basic building blocks -- $d(x^n)/dx=nx^{n-1}$ for

any positive integer $n$ (proved from first principles via the binomial expansion of

$(x+h)^n$) and $d(\sin x)/dx=\cos x$ (proved from first principles using the sum-to-product

identity for $\sin(x+h)-\sin x$ together with the standard limit $\sin\theta/\theta\to1$) --

lets any polynomial be differentiated term by term, and any product or quotient built from

polynomial and trigonometric pieces be differentiated by combining the power rule, …