How Do You Find the Derivative of Rational Functions?

TL;DR

To find the derivative of rational functions, rewrite them using the power rule and chain rule. For example, the derivative of 1/x is -1/x^2, and for 1/(x+5), it is -1/(x+5)^2. Use the quotient rule for expressions like (5-9x)/(x^2+4), resulting in (9x^2-10x-36)/(x^2+4)^2.

Transcript

in this lesson we're going to talk about how to find the derivative of a rational function so let's differentiate 1 over x what is the derivative of 1 over x well first we need to rewrite the expression so this is the same as x to the minus one and now we could use the power rule and recall that the power rule the derivative of x to the n is n x ra... Read More

Key Insights

• ✊ The power rule, which states that the derivative of x^n is n*x^(n-1), is useful in finding the derivative of rational functions.
• 📏 Rational functions involve quotients of polynomials, and the derivative can be determined by applying the power rule and chain rule.
• 📏 The chain rule is necessary when differentiating rational functions where the denominator contains a function of x.
• 📏 The quotient rule is used when finding the derivative of a rational function with both the numerator and denominator involving variables.
• 😑 Differentiating rational functions often requires simplifying the expressions and combining like terms before obtaining the final derivative.

Questions & Answers

Q: How do you find the derivative of 1/x?

To find the derivative of 1/x, you can rewrite it as x^-1 and apply the power rule to get -1/x^2.

Q: What is the derivative of 1/(x+5)?

The derivative of 1/(x+5) can be found by using the power rule and chain rule, resulting in -1/(x+5)^2.

Q: How do you differentiate 1/(4x-7)?

To differentiate 1/(4x-7), use the power rule and chain rule. The derivative is -4/(4x-7)^2.

Q: What is the derivative of 9/(x^2-6)?

The derivative of 9/(x^2-6) is -18x/(x^2-6)^2. Apply the power rule and chain rule to find the derivative.

Q: How do you find the derivative of 12/(4-8x)^4?

To find the derivative of 12/(4-8x)^4, use the power rule and chain rule. The derivative is 384/(4-8x)^5.

Q: How do you differentiate (5-9x)/(x^2+4)?

To differentiate (5-9x)/(x^2+4), use the quotient rule. The derivative is (9x^2-10x-36)/(x^2+4)^2.

Q: What is the derivative of (5-2x)/(x+4)^5?

The derivative of (5-2x)/(x+4)^5 can be found by using a combination of the chain rule and the quotient rule. The derivative is -65(5-2x)^4/(x+4)^6.

Summary & Key Takeaways

• To find the derivative of 1/x, rewrite it as x^-1 and apply the power rule: n(x^n-1), resulting in -1/x^2.

• For 1/(x+5), use the power rule and chain rule: -1/(x+5)^2.

• To differentiate 1/(4x-7), use the power rule and chain rule: -4/(4x-7)^2.

• Apply the power rule and chain rule to find the derivative of 9/(x^2-6): -18x/(x^2-6)^2.

• Use the power rule and chain rule to find the derivative of 12/(4-8x)^4: 384/(4-8x)^5.

• Apply the quotient rule to differentiate (5-9x)/(x^2+4): (9x^2-10x-36)/(x^2+4)^2.

• Use a combination of the chain rule and the quotient rule to find the derivative of (5-2x)/(x+4)^5: -65(5-2x)^4/(x+4)^6.