Understanding Derivatives and Chain Rule

The video tutorial explains how to find the derivative of a function h(x) = sqrt(75 + f(x)) using the power and chain rules. It starts by rewriting the square root as a rational exponent, then applies differentiation rules to find h'(x). The derivative is simplified using positive exponents, and the value of h'(3) is calculated by substituting given values for f(3) and f'(3). The final result is h'(3) = 7/18.

What is the initial expression given for h(x)?

• h(x) = 75 + f(x)

• h(x) = sqrt(75 + f(x))

• h(x) = 75 * f(x)

• h(x) = (75 + f(x))^2

What is the value of f(3) given in the problem?

• 7

• 6

• 8

• 5

How can the square root of a function be expressed using exponents?

• As a power of 2

• As a power of 1/2

• As a power of 3/2

• As a power of 1

What rule is applied to differentiate h(x) after rewriting it?

• Power Rule with Chain Rule

• Quotient Rule

• Product Rule

• Sum Rule

What is the inner function u in the expression for h(x)?

• u = 75 + f(x)

• u = f(x)

• u = 75

• u = 75 * f(x)

What is the derivative of the inner function u, denoted as u'?

• u' = 0

• u' = f'(x)

• u' = 75

• u' = 1

How is h'(x) expressed after applying the chain rule?

• h'(x) = f'(x) / (2 * sqrt(75 + f(x)))

• h'(x) = 2 * f'(x) * sqrt(75 + f(x))

• h'(x) = f'(x) * (75 + f(x))^2

• h'(x) = 2 * f'(x) / sqrt(75 + f(x))

What is the value of f'(3) used in the final calculation?

• 5

• 6

• 7

• 8

What is the final value of h'(3) after substitution and calculation?

• 1/2

• 7/18

• 9/18

• 7/9

What is the square root of 81, which is used in the final calculation?

• 10

• 9

• 8

• 7