Understanding Derivatives and Power Rule

The video tutorial explains how to find the derivative of the function H(x) = 5x^(1/4) + 7 and evaluate it at x = 16. It begins by introducing the problem and then demonstrates how to take the derivative using the power rule, even with a fractional exponent. The tutorial simplifies the derivative expression and evaluates it at the given point, x = 16, resulting in a final answer of 5/32.

What is the function H(x) given in the problem?

• H(x) = 5x^1/4 + 7

• H(x) = 5x^3 + 7

• H(x) = 5x^1/2 + 7

• H(x) = 5x^2 + 7

What is the first step in finding the derivative of H(x)?

• Divide by x

• Add a constant

• Apply the power rule

• Multiply by the constant

What is the purpose of using the power rule in this problem?

• To find the integral

• To simplify a fraction

• To solve an equation

• To find the derivative

What is the derivative of a constant with respect to x?

• 1

• x

• 0

• 7

How do you simplify the expression 5 * 1/4 x^(1/4 - 1)?

• 5/4 x^(3/4)

• 5/4 x^(-3/4)

• 5/4 x^(1)

• 5/4 x^(1/4)

What is the value of 16^(1/4)?

• 2

• 4

• 16

• 8

How is 16^(3/4) calculated?

• Take the square root of 16 and square it

• Cube 16 and take the fourth root

• Take the fourth root of 16 and cube it

• Square 16 and take the cube root

What is the final value of H'(16)?

• 5/16

• 5/4

• 5/32

• 5/8

What operation is performed after finding 16^(1/4) to evaluate H'(16)?

• Add 7

• Multiply by 5

• Cube the result

• Divide by 4

What is the result of multiplying 5 by 1/8 in the final calculation?

• 5/16

• 5/32

• 5/8

• 5/4