skip to contentstylized crane logoWhite Crane Education
Lectures>Practice

Exercises

Icon Legend

Exercises

Find the derivatives of the following functions.

  1. $f(x) = \ln(x^2 + 1)$green A - final answer
  1. $e^{xy+1}=y$
  1. $g(x) = \log_3(x^2 + 1)$
  1. green star - important content $e^{\ln(x+y+1)} = x$green question mark - hintgreen check mark - show solutiongreen A - final answer
  1. $f(x) = e^{x^2 + 1}$green A - final answer
  1. $\ln(x+y)=x+y+1$
  1. $g(x) = 3^{x^2 + 1}$
  1. $h(x) = x\ln(3x+1)$green A - final answer
  1. $f(x) = 4\log(e^x+1)$
  1. $f(x) = e^{\cos x}$green A - final answer
  1. $f(x) = e^{(x+1)^2}$
  1. $f(x) = x^2 + 4^{2x-1}$green A - final answer
  1. $g(x) = \ln\ln x$
  1. $m(x) = \cos(\ln x)$green A - final answer
  1. $t(x) = \frac{e^x}{e^x+1}$
  1. $f(x) = \frac{\ln x}{x + 1}$green A - final answer
  1. $g(x) = \ln(\sqrt{x+1} + x)$
  1. $m(x) = \ln \frac{x+a}{x-a}$green A - final answer
  1. Find a formula for the nth derivative of $f(x) = \ln x$.green check mark - show solution
  2. Find a formula for the nth derivative of $g(x) = e^{2x}$.green A - final answer
  3. Find a formula for the nth derivative of $h(x) = xe^{x}$.green check mark - show solution

Explorations

Logarithmic differentiation is a method where you start by taking the logarithm of a function, simplify it using logarithm properties then differentiate it implicitly. Thie method is particularly useful in situations where there are variables both in the base of an expression and in its exponent. Use this procedure to differentiate the following functions.

  1. $y = x^x$green check mark - show solution
  1. $y = x^{\cos x}$green check mark - show solution
  1. $y = (\sqrt{x})^{x}$
  1. green star - important content $y = \sqrt{x}$green question mark - hintgreen check mark - show solution
  1. $y = (x^2 + 1)^2(x^3 - 1)^4$green check mark - show solution

  1. $y = x^x$

Icons courtesy of icons8.com


link to Facebook