Text Differentiate the following composite functions: (a) y=(x3+1)5y=(x^{3}+1)^{5}y=(x3+1)5 (b) y=(x2−7x+6)4y=(x^{2}-7x+6)^{4}y=(x2−7x+6)4 (c) y=x2+9y=\sqrt{x^{2}+9}y=x2+9 Solution Chain rule (un)′=n un−1u′\left(u^{n}\right)'=n\,u^{n-1}u'(un)′=nun−1u′. (a) y′=5(x3+1)4⋅3x2=15x2(x3+1)4y'=5(x^{3}+1)^{4}\cdot 3x^{2}=15x^{2}(x^{3}+1)^{4}y′=5(x3+1)4⋅3x2=15x2(x3+1)4. (b) y′=4(x2−7x+6)3(2x−7)y'=4(x^{2}-7x+6)^{3}(2x-7)y′=4(x2−7x+6)3(2x−7). (c) y=(x2+9)1/2⇒y′=xx2+9y=(x^{2}+9)^{1/2}\Rightarrow y'=\dfrac{x}{\sqrt{x^{2}+9}}y=(x2+9)1/2⇒y′=x2+9x.