Differentiation and Derivative Concepts

This video is a continuation of implicit differentiation, focusing on examples with transcendental functions. It reviews the steps of implicit differentiation and provides multiple examples, including trigonometric, logarithmic, and exponential functions. The video also demonstrates solving for dy/dx and finding the slope of a tangent line at a specific point.

What is the first step in implicit differentiation?

• Differentiate both sides with respect to y

• Apply the product rule

• Differentiate both sides with respect to x

• Isolate dy/dx on one side

In the first example, what rule is applied to differentiate the term x * tangent y?

• Quotient Rule

• Chain Rule

• Product Rule

• Power Rule

What is the derivative of sine x with respect to x?

• Secant x

• Sine x

• Cosine x

• Tangent x

In the second example, what is the derivative formula for natural log u?

• u' * ln(u)

• 1/u

• 1/u * u'

• u * ln(u)

How do you simplify the expression for dy/dx in the second example?

• Combine like terms

• Apply the quotient rule

• Write as separate fractions

• Use the chain rule

In the third example, what is the derivative of cosine y with respect to x?

• -sine y * dy/dx

• -cosine y * dy/dx

• cosine y

• sine y

What is the result of dividing a negative by a negative in the third example?

• Undefined

• Positive

• Zero

• Negative

In the final example, what is the derivative of e^u with respect to x?

• u * e^u

• u'

• e^u * u'

• e^u

What is the slope of the tangent line at the point (3,0) in the final example?

• 9

• 3

• 1/9

• 1/3

What is the value of e^0?

• Undefined

• e

• 1

• 0