Radical functions differentiation | Derivative rules | AP Calculus AB | Khan Academy

TL;DR

The video explains how to use the chain rule to find the derivative of the fourth root of an expression and provides an example of finding the slope of the tangent line.

Transcript

• [Voiceover] Let's see if we can take the derivative with respect to x of the fourth root of x to the third power plus four x squared plus seven. And at first you might say, "All right, "how do I take the derivative of the fourth root "of something, it looks like I have a composite function, "I'm taking the fourth root of another expression?" And ... Read More

Key Insights

• 😑 The fourth root of an expression can be represented as a fractional exponent.
• 🫚 The chain rule is applied when taking the derivative of the fourth root function.
• ✊ The power rule is used to find the derivative of the outer function.

Questions & Answers

Q: How can the fourth root of an expression be rewritten as a fractional exponent?

The fourth root of an expression can be represented as the expression raised to the power of 1/4. For example, the fourth root of x is equivalent to x^(1/4).

Q: What is the chain rule used for in calculus?

The chain rule is used to find the derivative of a composite function. It allows us to differentiate the outer function with respect to the inner function and then multiply it by the derivative of the inner function.

Q: How do you find the derivative of the outer function using the power rule?

To find the derivative of the outer function, multiply the exponent by the coefficient and subtract 1 from the original exponent. For example, if the exponent is 1/4, the derivative would be (1/4) * (x^(1/4 - 1)).

Q: In the example given, what is the slope of the tangent line for the function y = fourth root of (x^3 + 4x^2 + 7) when x is equal to -3?

The slope of the tangent line is found by substituting x = -3 into the derivative equation and simplifying the expression. The resulting slope is 3/32.

Summary & Key Takeaways

• To find the derivative of the fourth root of an expression, rewrite it as a fractional exponent.

• Treat the fourth root as an outer function and the expression inside as an inner function.

• Use the power rule to find the derivative of the outer function and multiply it by the derivative of the inner function.

• An example is given to find the slope of the tangent line for the function y = fourth root of (x^3 + 4x^2 + 7), when x is equal to -3.