Chain rule for derivative of 2^x

TL;DR

The derivative of 2 to the x power can be simplified as the natural log of 2 times 2 to the x.

Transcript

Let's see if we can take the derivative with respect to x of 2 to the x power. And you might say, hold on a second. We know how to take the derivative of e to the x. But what about a base like 2? We don't know what to do with 2. And the key here is to rewrite 2 to the x so that we essentially have it as e to some power. And the key there is to rewr... Read More

Key Insights

• ✊ The derivative of 2 to the x power can be found by rewriting it as e to the natural log of 2 raised to the x power.
• 😑 By using the chain rule and differentiating the exponential function, we can simplify the derivative expression.
• 🧑‍💻 The simplified derivative is the natural log of 2 times 2 to the x.
• 👻 The application of properties of e and the chain rule allows for an easier computation of the derivative.
• 😑 Understanding the relationship between 2, e, and the natural log of 2 helps in simplifying the expression.
• 📏 Differentiation rules and properties of exponents are utilized to find the derivative.
• 😑 The exponential function e to the x plays a crucial role in simplifying the derivative expression.

Questions & Answers

Q: How can we rewrite 2 to the x as e to some power?

We can rewrite 2 as e to the natural log of 2 and raise it to the x power. This allows us to use the properties of e, making it easier to find the derivative.

Q: What is the derivative of e to the something with respect to that something?

The derivative of e to the something with respect to that something is simply e to that something. It follows from the definition of the exponential function.

Q: How do we find the derivative of the natural log of 2 times x?

The derivative of the natural log of 2 times x with respect to x is just the coefficient on x, which is the natural log of 2. Differentiating the product of a constant and x gives us the constant.

Q: How is the derivative of 2 to the x simplified?

The derivative of 2 to the x is simplified as the natural log of 2 times 2 to the x. This is obtained by rewriting the expression using properties of e and applying the chain rule.

Summary & Key Takeaways

• To find the derivative of 2 to the x power, rewrite it as e to the natural log of 2 raised to the x power.

• Utilize the chain rule by taking the derivative of e to the natural log of 2 times x with respect to the natural log of 2 and then multiplying it by the derivative of the natural log of 2 times x with respect to x.

• Simplify the expression to the natural log of 2 times 2 to the x.