Differentiation Techniques and Rules

This video tutorial covers the process of finding derivatives of rational functions. It begins with an introduction to the power rule and chain rule, followed by several examples demonstrating how to apply these rules to differentiate various rational functions. The tutorial includes step-by-step solutions for functions like 1/x, 1/(x+5), 1/(4x-7), 9/(x^2-6), 12/(4-8x)^4, and (5-9x)/(x^2+4), emphasizing the use of power, chain, and quotient rules.

What is the derivative of 1/x using the power rule?

• x^2

• 1/x^2

• -1/x^2

• -x^2

How do you rewrite 1/(x+5) to apply the power rule?

• (x+5)^2

• (x+5)^-1

• (x+5)^0

• (x+5)^1

What is the derivative of 1/(4x-7) using the chain rule?

• -4/(4x-7)^2

• -1/(4x-7)^2

• 1/(4x-7)^2

• 4/(4x-7)^2

Which rule is used to differentiate a function with variables in both the numerator and denominator?

• Quotient rule

• Chain rule

• Product rule

• Power rule

What is the derivative of (5-9x)/(x^2+4) using the quotient rule?

• (x^2+4)(-9) - (5-9x)(2x) / (x^2+4)^2

• (x^2+4)(9) + (5-9x)(2x) / (x^2+4)^2

• (x^2+4)(9) - (5-9x)(2x) / (x^2+4)^2

• (x^2+4)(-9) + (5-9x)(2x) / (x^2+4)^2

What is the derivative of 12/(4-8x)^4 using the power rule?

• -48/(4-8x)^5

• 48/(4-8x)^5

• 384/(4-8x)^5

• -384/(4-8x)^5

What is the derivative of 9/(x^2-6) using the chain rule?

• -18x/(x^2-6)^2

• 18x/(x^2-6)^2

• -9x/(x^2-6)^2

• 9x/(x^2-6)^2

How do you start differentiating (5-2x)/(x+4)^5 using the chain rule?

• Divide the exponent by the inside function

• Multiply the exponent by the inside function

• Add the exponent to the front and subtract one from it

• Move the exponent to the front and subtract one from it

What is the result of applying the chain rule to (5-2x)/(x+4)^5?

• 5(5-2x)^4/(x+4)^5

• -5(5-2x)^4/(x+4)^5

• 5(5-2x)^5/(x+4)^4

• -5(5-2x)^5/(x+4)^4

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

• 65(5-2x)^5/(x+4)^6

• -65(5-2x)^5/(x+4)^6

• 65(5-2x)^4/(x+4)^6

• -65(5-2x)^4/(x+4)^6