Differentiation Rules and Applications

The video tutorial explains how to calculate the derivative of a function k(x) using different rules of differentiation. It covers three parts: Part A involves finding k'(20) using the basic derivative rules for a linear combination of functions. Part B applies the product rule to determine k'(20) when k(x) is a product of two functions. Part C uses the quotient rule to find k'(20) when k(x) is a quotient of two functions. The tutorial provides step-by-step calculations and simplifies the results for each part.

What is the derivative of a function that is a linear combination of two other functions?

• The sum of the derivatives of the two functions

• The product of the derivatives of the two functions

• The difference of the derivatives of the two functions

• The quotient of the derivatives of the two functions

If f'(20) = 17 and g'(20) = -11, what is the value of k'(20) for k(x) = -8f(x) + 13g(x)?

• -427

• -143

• -136

• -279

Which rule is used to differentiate the product of two functions?

• Quotient Rule

• Power Rule

• Product Rule

• Chain Rule

For k(x) = f(x)g(x), what is the expression for k'(x) using the product rule?

• f(x)g'(x) + g(x)f'(x)

• f(x)g(x) + f'(x)g'(x)

• f'(x)g'(x)

• f'(x)g(x) - f(x)g'(x)

Given f(20) = 11, g'(20) = -11, g(20) = -18, and f'(20) = 17, what is k'(20) for k(x) = f(x)g(x)?

• -427

• -306

• -121

• -185

Which rule is applied to differentiate a quotient of two functions?

• Chain Rule

• Quotient Rule

• Product Rule

• Power Rule

For k(x) = f(x)/g(x), what is the expression for k'(x) using the quotient rule?

• (f(x)g(x)) / g(x)^2

• (f(x)g'(x) + g(x)f'(x)) / g(x)^2

• (g(x)f'(x) - f(x)g'(x)) / g(x)^2

• (f'(x)g(x) - g'(x)f(x)) / f(x)^2

If g(20) = -18, f'(20) = 17, f(20) = 11, and g'(20) = -11, what is k'(20) for k(x) = f(x)/g(x)?

• -306/324

• -185/324

• -121/324

• -427/324

What is the denominator in the quotient rule for k(x) = f(x)/g(x)?

• g(x)

• f(x)

• g(x)^2

• f(x)^2

What is the result of simplifying the expression for k'(20) using the quotient rule given the function values?

• -185/324

• -306/324

• -121/324

• -427/324