Derivatives of Exponential Functions & Logarithmic Differentiation Calculus lnx, e^2x, x^x, x^sinx

The Organic Chemistry Tutor・2 minutes read

The text covers various examples and rules for finding derivatives, including the application of the power rule, product rule, and chain rule, as well as specific derivatives for exponential and logarithmic functions. It demonstrates how to handle different types of functions and constants when finding derivatives through various rules and formulas.

Insights

  • Derivative rules for exponential functions show that the derivative of e to the 2x is 2e to the 2x, and the derivative of e to the 3x is 3e to the 3x, demonstrating how to differentiate exponential functions efficiently using the formula e to the u times the derivative of u.
  • Applying logarithmic and exponential differentiation techniques, the derivative of complex functions like x to the x is x^x(ln x + 1), showcasing the power of logarithmic and chain rules in handling intricate derivatives involving exponential and logarithmic functions.

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Recent questions

  • What is the derivative of e to the 2x?

    2e to the 2x

  • How do you find the derivative of x^3?

    3x^2

  • What is the derivative of ln x?

    1/x

  • How do you find the derivative of x ln x?

    ln x + 1

  • What is the derivative of x to the x?

    x^x(ln x + 1)

Related videos

Summary

00:00

Differentiation Rules for Exponential and Logarithmic Functions

  • Derivative of e to the 2x is 2e to the 2x, following the formula e to the u times the derivative of u.
  • Power rule states that the derivative of x^n is n times x^(n-1), exemplified by derivatives of x^3, x^4, and x^5.
  • Derivative of 7x^6 is 42x^5, showcasing how to handle a number in front of x in differentiation.
  • Derivative of x is 1, and for 3x it's 3, for -4x it's -4, and for 2x it's 2, with an invisible 1 for x.
  • Derivative of 2x is 2, proven using the power rule by finding the derivative of x^1.
  • Derivative of a constant like 8 is 0, and for e without x it's 0, while e^x is e^x times 1.
  • Derivative of x^e is e*x^(e-1), applying the power rule for a variable raised to a constant.
  • Derivative of e to the 3x is 3e to the 3x, calculated using the formula e to the u times u prime.
  • Derivative of ln x is 1/x, derived from the formula for the derivative of ln u as u prime over u.
  • Derivative of ln x^2 is 2/x, obtained by simplifying the expression and applying the derivative formula for ln u.

19:04

Derivatives of Logarithmic and Exponential Functions

  • Derivative of 2x is 2, base a is 3, formula for derivative is u'/(u*ln(a)), simplifying gives 1/(x*ln(3)).
  • For log base 5 of x squared, derivative of x squared is 2x, base a is 5, applying formula gives 2/(x*ln(5)).
  • Derivative of log base 7 of 1-x, derivative of -x is -1, base a is 7, result is -1/(1-x*ln(7)).
  • Derivative of x ln x using product rule, result is ln x + 1.
  • Derivative of x squared ln x using product rule, result is 2ln x + 1.
  • Derivative of x e to the x using product rule, result is e to the x (1 + x).
  • Derivative of ln ln x is 1/(x*ln x).
  • Derivative of ln ln ln x is 1/(x*ln x*ln x).
  • Derivative of ln x/x is 1 - ln x/x^2.
  • Derivative of x to the x is x^x(ln x + 1) using logarithmic differentiation.

40:39

Derivative of ln y equals x^(ln x)

  • Taking the natural log of both sides, then moving the exponent to the front, results in ln y = ln x * ln x. The derivative of ln y is 1/y * dy/dx, which can be found using the product rule to get 2 ln x / x or by writing ln x * ln x as ln x^2 and applying the chain rule to get the same result. Finally, multiplying y by both sides and replacing y yields dy/dx = x^(ln x) * 2 ln x / x.
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