Understanding the Derivative of Natural Logarithm

This video tutorial provides a proof of the derivative of the natural logarithm function, ln(x), using implicit differentiation. The process begins by setting y equal to ln(x) and converting it to an exponential form. The tutorial then differentiates both sides of the equation with respect to x, applying the chain rule. Finally, it solves for dy/dx, demonstrating that the derivative of ln(x) is 1/x. The video concludes with a summary of the proof.

What is the main goal of using implicit differentiation in this proof?

• To differentiate a polynomial function

• To solve for X in terms of Y

• To prove the derivative of natural log X is 1/X

• To find the derivative of e^x

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What is the equivalent exponential form of the equation natural log X = Y?

• e^X = Y

• X^Y = e

• Y^e = X

• e^Y = X

Why do we apply the chain rule when differentiating e^Y with respect to X?

• Because Y is a constant

• Because e^Y is a polynomial

• Because X is a function of Y

• Because Y is a function of X

What additional factor is introduced when applying the chain rule to e^Y?

• dX/dt

• dY/dX

• dX/dY

• dY/dt

What is the derivative of X with respect to X?

• X

• 0

• 1

• e^X

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How do we solve for dY/dX in the equation e^Y * dY/dX = 1?

• Add e^Y to both sides

• Multiply both sides by e^Y

• Subtract e^Y from both sides

• Divide both sides by e^Y

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What substitution is made to express dY/dX in terms of X?

• e^Y = X

• e^Y = Y

• e^Y = e

• e^Y = 1

What is the final result of the proof for the derivative of natural log X?

• dY/dX = 1/X

• dY/dX = e^X

• dY/dX = 0

• dY/dX = X