The Derivative of Exponential and Logarithmic Functions (with examples)

• Example 1 - Determine the first derivative of a function \(y = x^7 e^x\)
  Solution- \begin{eqnarray}y' &=& 7x^6e^x+x^7 e^x \\\nonumber y' &=& e^xx^6(7+x)\end{eqnarray}
• Example 2 - Determine tthe frist derivative of a function \(y = (x-1) e^x\)
  Solution \begin{eqnarray}y' &=& e^x + xe^x - e^x \\\nonumber y' &=& xe^x\end{eqnarray}
• Example 3 Determine the first derivative of a function \(y = \frac{e^x}{x^2}\)
  Solution \begin{eqnarray}y' &=& \frac{e^x x^2 - 2xe^x}{x^4} \\\nonumber y' &=& \frac{xe^x(x-2)}{x^4}\\\nonumber y' &=& \frac{e^x(x-2)}{x^3}\end{eqnarray}
• Example 4 Determine the first derivative of a function \(y = \frac{x^5}{e^x}\)
  Solution \begin{eqnarray}y'&=& \frac{5x^4 e^x - x^5 e^x}{(e^x)^2}\\\nonumber y'&=& \frac{e^xx^4(5-x)}{e^{2x}} \\\nonumber y'&=& \frac{x^4(5-x)}{e^x}\end{eqnarray}
• Example 5 Determine the first derivative of a function \(y = e^x \cos x\)
  Solution \begin{eqnarray}y' &=& e^x\cos x - e^x \sin x \end{eqnarray}
• Example 6 Determine the first derivative of a function \(y = (x^2 -2x+2)e^x\)
  Solution \begin{eqnarray}y'&=& 2xe^x + x^2 e^x - 2e^x -2xe^x 2e^x\\\nonumber y' &=& e^x(2x+x^2-2-2x +2 ) \\\nonumber y'&=& x^2e^x\end{eqnarray}
• Example 7 Determine the first derivative of a function \(y = e^x\mathrm{arcsin} x\)
  Solution \begin{eqnarray}y'&=& e^x \mathrm{arcsin} x + \frac{e^x}{\sqrt{1-x^2}}\end{eqnarray}
• Example 8 Determine the first derivative of a function \(y = \frac{x^2}{\ln x}\)
  Solution \begin{eqnarray}y'&=& \frac{2x\ln x - x^2 \frac{1}{x}}{(\ln x)^2}\\\nonumber y' &=& \frac{2x\ln x - x}{\ln^2 x}\end{eqnarray}
• Example 9 Determine the first derivative of a function \(y = x^3\ln x - \frac{x^3}{3}\)
  Solution \begin{eqnarray}y' &=& 3x^2\ln x + x^3 \frac{1}{x} - \frac{3x^2}{3} \\\nonumber y'&=& 3x^2\ln x + x^2 - x^2 \\\nonumber y'&=& 3x^2\ln x\end{eqnarray}
• Example 10 Determine the first derivative of a function \(y = \frac{1}{x} + 2\ln x -\frac{\ln x}{x}\)
  Solution \begin{eqnarray}y' &=& -\frac{1}{x^2} +\frac{2}{x} - \frac{\frac{1}{x}\cdot x - \ln x}{x^2}\\\nonumber y' &=& -\frac{1}{x^2} + \frac{2}{x} - \frac{1-\ln x}{x^2}\end{eqnarray}