Q.Differentiate $y=4x^5-3x^3+2x-9$.

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,

$d(\cos x)/dx=-\sin x$ (proved the same way), and the product/quotient rules. The remaining

trigonometric derivatives, such as $d(\tan x)/dx=\sec^2x$, follow from these two base results by

writing the function as a quotient (e.g. $\tan x=\sin x/\cos x$) and applying the quotient rule,

rather than needing a fresh first-principles proof of their own.