Derivatives of composite functions (roots)

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Differentiate using the chain rule ($a$ constant):

• (a) $y=\sqrt{x^{2}+1}$
• (b) $y=\dfrac{1}{\sqrt{a+x}}$
• (c) $y=\dfrac{\sqrt{x^{2}-a^{2}}}{x^{2}}$
• (d) $y=\dfrac{a^{2}+x^{2}}{(a+x)^{2}}$

Solution

• (a) $y'=\dfrac{2x}{2\sqrt{x^{2}+1}}=\dfrac{x}{\sqrt{x^{2}+1}}$
• (b) $y=(a+x)^{-1/2}\Rightarrow y'=-\dfrac{1}{2}(a+x)^{-3/2}=-\dfrac{1}{2\sqrt{(a+x)^{3}}}$
• (c) differentiating $x^{-2}\sqrt{x^{2}-a^{2}}$ with product and chain rules: $y'=\frac{1}{x\sqrt{x^{2}-a^{2}}}-\frac{2\sqrt{x^{2}-a^{2}}}{x^{3}}=\frac{x^{2}-2(x^{2}-a^{2})}{x^{3}\sqrt{x^{2}-a^{2}}}=\frac{2a^{2}-x^{2}}{x^{3}\sqrt{x^{2}-a^{2}}}.$
• (d) quotient: $y'=\dfrac{2x(a+x)^{2}-(a^{2}+x^{2})\cdot2(a+x)}{(a+x)^{4}}=\dfrac{2a(x-a)}{(a+x)^{3}}$ $\boxed{\tfrac{x}{\sqrt{x^{2}+1}};\ -\tfrac{1}{2\sqrt{(a+x)^{3}}};\ \tfrac{2a^{2}-x^{2}}{x^{3}\sqrt{x^{2}-a^{2}}};\ \tfrac{2a(x-a)}{(a+x)^{3}}}$