DIFFERENTIATION USING THE CHAIN RULE
The following problem requires the use of the chain rule. The chain rule is a rule for differentiating compositions of functions. In the following discussion and solution the derivative of a function h(x) may be denoted by h'(x) . The chain rule states formally that
\[ \frac{d}{dx}f(g(x)) = (\frac{d}{dx}(f(x)))(g(x)) \frac{d}{dx}g(x) \]
Differentiate

\[y = {\sqrt{13x^2  5x +8}}\]

Solution:
Differentiate the outer layer first, the square root.
\[d/dx {\sqrt{13x^2  5x +8}}\]
\[d/dx ({13x^2  5x +8})^{1/2}\]
\[(1/2) ({13x^2  5x +8})^{1/2  1} d/dx ({13x^2  5x +8})\]
\[(1/2) ({13x^2  5x +8})^{1/2} (26x  5)\]
\[\frac{26x  5}{2(13x^2  5x +8)^{1/2}}\]
\[\frac{26x  5}{2\sqrt{13x^2  5x +8)}}\]