The Derivative of Various Functions (with examples)

• Example 1 Determine the first derivative of a function \(y = \left(\frac{ax + b}{c}\right)^3\)
  Solution \begin{eqnarray}y' &=& 3 \left(\frac{ax+b}{x}\right)^2 \left(\frac{a}{c}\right)\end{eqnarray}
• Example 2 Determine the first derivative of a function \(y = (2a+ 3bx)^2\)
  Solution \begin{eqnarray}y &=& (2a+ 3bx)^2\\\nonumber y'&=& 2(2a+3bx)(3b) \end{eqnarray}
• Example 3 Determine the first derivative of a function \(y = \frac{3}{56(2x-1)^7} - \frac{1}{24(2x-1)^6} - \frac{1}{40(2x-1)^5}\)
  Solution \begin{eqnarray}y &=& \frac{3}{56(2x-1)^7} - \frac{1}{24(2x-1)^6} - \frac{1}{40(2x-1)^5}\\\nonumber y' &=& -\frac{3}{4(2x-1)^7}+ \frac{1}{2(2x-1)^7} + \frac{1}{4(2x-1)^6} \\\nonumber y'&=& \frac{-3 + 2(2x-1)+4x^2 - 4x + 1}{4(2x-1)^8} \\\nonumber y'&=& \frac{-3 + 4x - 2 + 4x^2 - 4x +1 }{4(2x-1)^8} \\\nonumber y' &=& \frac{4(x^2-1)}{4(2x-1)^8} \\\nonumber y'&=& \frac{x^2 -1}{(2x-1)^8}\end{eqnarray}
• Example 4 Determine the first derivative of the function \(y = \sqrt{1-x^2}\)
  Solution \begin{eqnarray}y &=& \sqrt{1-x^2}\\\nonumber y' &=& \frac{1}{2\sqrt{1-x^2}}(-2x) \\\nonumber y' &=& -\frac{x}{\sqrt{1-x^2}}\end{eqnarray}
• Example 5 Determine the first derivative of the function \(y = \sqrt[3]{a + bx^3}\)
  Solution \begin{eqnarray}y &=& \sqrt[3]{a + bx^3} \\\nonumber y'&=& \frac{1}{3}(a + bx^3)^{\frac{1}{3} -1} (3bx^2) \\\nonumber y' &=& \frac{1}{3} (a + bx^3)^{-\frac{2}{3}} (3bx^2) \\\nonumber y' &=& \frac{bx^2}{\sqrt[3]{(a+bx^3)^2}}\end{eqnarray}
• Example 6 Determine the first derivative of function \(y = \left(a^{\frac{2}{3}} - x^{\frac{2}{3}}\right)^{\frac{3}{2}}\)
  Solution \begin{eqnarray}y &=& \left(a^{\frac{2}{3}} - x^{\frac{2}{3}}\right)^{\frac{3}{2}} \\\nonumber y' &=& \frac{3}{2}\left(a^{\frac{2}{3}} - x^{\frac{2}{3}}\right)^{\frac{3}{2}} \left(-\frac{2}{3}x^{\frac{2}{3}-1}\right) \\\nonumber y' &=& -\frac{\sqrt{(a^{\frac{2}{3}} - x^{\frac{2}{3}})}}{\sqrt[3]{x}}\end{eqnarray}
• Example 7 Determine the first derivative of a function \(y = (3-2\sin x)^5\)
  Solution \begin{eqnarray}y &=& (3-2\sin x)^5 \\\nonumber y' &=& 5(3-2\sin x)^4 (-2\cos x) \\\nonumber y' &=& -10 \cos x(3-2\sin x)^4\end{eqnarray}
• Example 8 Determine the first derivative of a function \(y = \sin^35x \cos^2 \frac{x}{3}\)
  Solution \begin{eqnarray}y' &=& 3\sin^2 5x \cos 5x \cdot 5 \cdot \cos^2 \frac{x}{3} + \sin^3 5x \cdot 2 \cos \frac{x}{3}\cdot \left(-\sin \frac{x}{3}\right)\cdot \frac{1}{3}\\\nonumber y' &=& 15 \cdot \sin^2 5x \cos 5x \cos^2 \frac{x}{3} -\frac{2}{3}\sin 5x \sin\frac{x}{3}\cos\frac{x}{3}\end{eqnarray}
• Example 9 Determine the first derivative of a function \(y = -\frac{11}{2(x-2)^2} - \frac{4}{x-2}\)
  Solution \begin{eqnarray}y' &=& \frac{1}{2(x-2)^3}(-2) - \frac{4}{(x-2)^2}(-1)\\\nonumber y'&=& -\frac{1}{(x-2)^3} + \frac{4}{(x-2)^2}\\\nonumber y'&=& \frac{-1 + 4(x-2)}{(x-2)^3} \\\nonumber y' &=& \frac{-1 + 4x-8}{(x-2)^3}\\\nonumber y'&=& \frac{4x-9}{(x-2)^3}\end{eqnarray}
• Example 10 Determine the first derivative of a function \(y = -\frac{15}{4(x-3)^4} - \frac{10}{3(x-3)^3} - \frac{1}{2(x-3)^2}\)
  Solution This function is little bit complicated than in previous examples so the first derivative of each function member will be made separately. First member: \begin{eqnarray}y_I &=& -\frac{15}{4(x-3)^4}\\\nonumber y_I'&=& -\frac{15}{4} (-4) (x-3)^{-4-1} = \frac{15}{(x-3)^5}\end{eqnarray} Second member: \begin{eqnarray}y_{II} &=& -\frac{10}{3(x-3)^3}\\\nonumber y_{II}' &=& -\frac{10}{3} (-3) (x-3)^{-3-1}\\\nonumber y_{II}' &=& \frac{10}{(x-3)^4}\end{eqnarray} Third member: \begin{eqnarray}y_{III} &=& -\frac{1}{2(x-3)^2}\\\nonumber y_{III}' &=& -\frac{1}{2}(-2) (x-3)^{-2-1} = \frac{1}{(x-3)^3}\end{eqnarray} Combining all derived members toghether the following solution is obtained \begin{eqnarray}y' &=& \frac{15}{(x-3)^5} + \frac{10}{(x-3)^4} + \frac{1}{(x-3)^3}\end{eqnarray}
• Example 11 Derive the first derivative of a function \(y = \frac{x^8}{8(1-x^2)^4}\)
  Solution \begin{eqnarray}y'&=& \frac{8x^7 (8(1-x^2)^4) - x^8 \cdot 8 \cdot 4(1-x^2)^3(-2x)}{64(1-x^2)^8}\\\nonumber y'&=& \frac{64 x^7(1-x^2)^4 + 64x^9(1-x^2)^3}{64(1-x^2)^3}\\\nonumber y'&=& \frac{64(1-x^2)^3x^7[(1-x^2) + x^2]}{64(1-x^2)^8}\\\nonumber y'&=& \frac{x^7}{(1-x^2)^8}\end{eqnarray}
• Example 12 Derive the first derivative of a function \(y = \frac{\sqrt{2x^2 -2x +1}}{x}\)
  Solution \begin{eqnarray}y &=& \frac{\sqrt{2x^2 -2x +1}}{x} \\\nonumber y'&=& \frac{\frac{1}{2}\frac{1}{\sqrt{2x^2-2x +1}}(4x-2)x - \sqrt{2x^2 -2x +1}}{x^2}\\\nonumber y'&=& \frac{\frac{2x^2-x}{\sqrt{2x^2-2x+1}} + \sqrt{2x^2-2x+1}}{x^2}\end{eqnarray}
• Example 13 Derive the first derivative of a function \(y=\frac{x}{\sqrt{a^2+x^2}}\)
  Solution \begin{eqnarray}y'&=& \frac{a^2 \sqrt{a^2+x^2} - x\frac{1}{2}a^2 \frac{2x}{\sqrt{a^2+x^2}}}{(a^2\sqrt{a^2+x^2})^2}\\\nonumber y'&=& \frac{a^2 \sqrt{a^2+x^2}-\frac{a^2 x^2}{\sqrt{a^2 +x^2}}}{a^4(a^2+x^2)} \\\nonumber y'&=& \frac{\frac{a^2 + x^2 - x^2}{\sqrt{a^2+x^2}}}{a^2(a^2+x^2)}\\\nonumber y'&=& \frac{1}{\sqrt{(a^2+x^2)^3}}\end{eqnarray}
• Example 14 Derive the first derivative of a function \(y = \frac{x^3}{3\sqrt{(1+x^2)^3}}\)
  Solution \begin{eqnarray}\end{eqnarray}
• Example 15 Derive the first derivative of a function \(y = \frac{3}{2}\sqrt[3]{x^2} + \frac{18}{7}x\sqrt[6]{x}+\frac{9}{5}x\sqrt[3]{x^2} + \frac{6}{13}x^2 \sqrt[6]{x}\)
  Solution \begin{eqnarray}y'&=& \frac{3}{2}\frac{2}{3}x^{\frac{2}{3}-1} + \frac{18}{7}\frac{7}{6}x^{\frac{7}{6}-1} + \frac{9}{5}\frac{5}{3}x^{\frac{5}{3}-1}+ \frac{6}{13}\frac{13}{6}x^{\frac{13}{6}-1} \\\nonumber y'&=& x^{-\frac{1}{3}} + 3x^{\frac{1}{6}} + 3x^{\frac{2}{3}} + x^{\frac{7}{6}}\end{eqnarray}
• Example 16 Derive the first derivative of a function \(y=\frac{1}{8}\sqrt[3]{(1+x^3)^8} - \frac{1}{5}\sqrt[3]{(1+x^3)^5}\)
  Solution \begin{eqnarray} y' &=& \frac{1}{8}\frac{8}{3}3x^2(1+x^3)^{\frac{8}{3}-1} - \frac{1}{5}\frac{5}{3}(1+x^3)^{\frac{5}{3}-1}3x^2 \\\nonumber y'&=& x^2(1+x^3)^{\frac{5}{3}} - x^2(1+x^3)^{\frac{2}{3}}\end{eqnarray}
• Example 17 Derive the first derivative of a function \(y =\frac{4}{3}\sqrt[4]{\frac{x-1}{x+2}}\)
  Solution \begin{eqnarray}y'&=& \frac{4}{3}\frac{1}{4}\left(\frac{x-1}{x+2}\right)^{\frac{1}{4}-1} \frac{(x+2)-(x-1)}{(x+2)^2}\\\nonumber y'&=& \frac{1}{3}\left(\frac{x-1}{x+2}\right)^{-\frac{3}{4}} \frac{x+2-x+1}{(x+2)^2}\\\nonumber y'&=& \frac{1}{\sqrt[4]{\left(\frac{x-1}{x+2}\right)^3} (x+2)^2} = \frac{1}{\sqrt[4]{(x-1)^3 (x+2)^5}}\end{eqnarray}