Derivative of sin(ln(x_)) | Advanced derivatives | AP Calculus AB | Khan Academy

TL;DR

Taking the derivative of the sine of the natural log of x squared involves applying the chain rule multiple times.

Transcript

So now we're going to attempt to take the derivative of the sine of the natural log of x squared. So now we have a function that's the composite of a function, that's a composite of another function. So one way you could think of it, if you set f of x as being equal to sine of x, and g of x being the natural log of x, and h of x equaling x squared.... Read More

Key Insights

• 📏 The given function involves composite functions, which require the application of the chain rule.
• 👨‍💼 The derivative of the sine function is cosine, but it is evaluated at the inner function.
• ❓ The derivative of the natural logarithm function is 1 over the argument of the function.
• ✖️ The derivative of a polynomial function like x squared is simply the coefficient multiplied by x.
• 😑 Simplifying the derived expression involves canceling out common factors and rearranging the terms.
• 🤪 The process of taking the derivative can be thought of as peeling an onion, going layer by layer.
• 📏 Understanding the chain rule is crucial in finding the derivative of composite functions.

Questions & Answers

Q: What is the first step in taking the derivative of the given function?

The first step is to apply the chain rule by taking the derivative of the outer function (sine) with respect to its inner function (natural log of x squared), which gives us cosine of the inner function.

Q: What is the next step after finding the derivative of the outer function?

The next step is to find the derivative of the inner function (natural log of x squared) with respect to x, which is 1/x squared.

Q: How do we find the derivative of the innermost function?

To find the derivative of the innermost function (x squared) with respect to x, we simply differentiate it, which gives us 2x.

Q: How can we simplify the derived expression?

We can simplify the derived expression by canceling out the common factors and rearranging the terms. In this case, we can cancel out the x and x squared, leaving us with 2/x times cosine of the natural log of x squared.

Summary & Key Takeaways

• The function we are trying to find the derivative of is a composite function, which can be broken down into individual functions.

• The derivative of the outer function (sine) with respect to its inner function (natural log of x squared) is cosine of the inner function.

• The derivative of the inner function (natural log of x squared) with respect to x is 1/x squared.

• The derivative of the innermost function (x squared) with respect to x is 2x.

• By applying the chain rule and simplifying, the final derivative is 2/x times cosine of the natural log of x squared.