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

TL;DR

This video explains how to find the derivative of exponential and logarithmic functions, including e^x, ln(x), and logarithmic differentiation, through step-by-step examples and rules.

Transcript

in this video we're going to focus on derivatives derivatives of exponential functions like e to the x logarithmic functions natural logs and also logarithmic differentiation like how do you differentiate x to the sine x and x raised to the x so we're going to go over all of that in this video so let's begin what is the derivative of e to the 2x go... Read More

Key Insights

• ⌛ The derivative of e^(ux) is e^(ux) times the derivative of u.
• 👻 The power rule allows us to find the derivative of functions with exponents.
• 🗂️ Derivatives of logarithmic functions involve dividing by the original function.
• 📏 Differentiation of functions with both exponential and logarithmic terms requires the product rule or quotient rule.

Questions & Answers

Q: What is the derivative of e^(2x)?

The derivative of e^(2x) is 2e^(2x), using the rule that the derivative of e^u is e^u times the derivative of u.

Q: How do you find the derivative of x^3?

To find the derivative of x^3, you use the power rule which states that the derivative of x^n is n times x^(n-1). Hence, the derivative of x^3 is 3x^2.

Q: What is the derivative of ln(x)?

The derivative of ln(x) is 1/x. This can be derived using the formula for the derivative of ln(u), which is u' / u, where u represents the expression inside the natural logarithm.

Q: How do you find the derivative of x^(sin(x))?

To find the derivative of x^(sin(x)), you use logarithmic differentiation. Take the natural log of both sides, then differentiate using the product rule and chain rule. The final result is (sin(x)*ln(x) + cos(x))/x * x^(sin(x)).

Summary & Key Takeaways

• The video covers the derivatives of exponential functions like e^x, logarithmic functions like ln(x), and logarithmic differentiation for functions such as x^sin(x) and x^x.

• The power rule is introduced, stating that the derivative of x^n is n times x^(n-1).

• Derivatives of exponential functions with constants in the exponent are explained.

• The product rule and quotient rule are used to find the derivatives of functions involving both exponential and logarithmic terms.

• Logarithmic differentiation is introduced as a method to find the derivative of functions with variable bases and exponents.