How to Differentiate the Fourth Root Using Chain Rule

TL;DR

To differentiate the fourth root of an expression, rewrite it as a fractional exponent and apply the chain rule. Calculate the derivative of the outer function, then multiply by the derivative of the inner function. The video provides a detailed example of finding the tangent slope at a specific point.

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.