derivative of x^y=y^x, implicit differentiation , calculus 1 tutorial, AP calculus | Summary and Q&A

TL;DR

The video explains the process of implicitly differentiating exponential functions using the natural logarithm and the product rule.

Key Insights

• 🙃 Implicit differentiation involves taking the natural logarithm on both sides of an equation to simplify the process.
• ⚾ The product rule is used to differentiate functions that involve both the base and the exponent.

Transcript

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Questions & Answers

Q: Why is it necessary to take the natural logarithm on both sides when implicitly differentiating exponential functions?

Taking the natural logarithm on both sides allows us to apply the log property that brings the exponent to the front. This simplifies the differentiation process and allows us to differentiate functions involving the base and the exponent separately.

Q: How is the product rule used in implicitly differentiating functions involving both the base and the exponent?

The product rule is used to differentiate functions that are a product of two functions. When differentiating the function y * ln(x), we keep the first function, y, and differentiate the second function, ln(x), using the derivative of ln(x) which is 1/x. Then, we add it with the second function, ln(x), multiplied by the derivative of y with respect to x.

Q: How is the derivative of y with respect to x, denoted as dy/dx, obtained in the differentiation process?

The derivative of y with respect to x is obtained by implicitly differentiating the equation and isolating the term dy/dx. By rearranging the equation, we obtain dy/dx = (ln(y) - y/x) / (x - x^2/y).

Q: How can the complex fraction in the final answer be simplified?

The complex fraction in the final answer can be simplified by multiplying the numerator and denominator by the common denominators of the small fractions involved. By multiplying both the numerator and denominator by xy, we can simplify the expression to xy ln(y) - y^2 / xy - x^2.

Summary & Key Takeaways

• Exponential functions can be differentiated using implicit differentiation by taking the natural logarithm on both sides of the equation.

• The product rule is used when differentiating functions that involve both the base and the exponent.

• The final result of the differentiation involves terms like ln(x), ln(y), x, and y.