Understanding Derivatives and Their Properties

The video tutorial covers various derivative rules, including the power rule, constant rule, and rules for trigonometric, exponential, and logarithmic functions. It provides a proof for the derivative of sine being cosine and demonstrates solving derivative problems using these rules.

What is the derivative of a constant function?

• Zero

• The constant itself

• One

• Undefined

If a function is multiplied by a constant, how does this affect its derivative?

• The derivative is zero

• The derivative is multiplied by the constant

• The derivative is divided by the constant

• The derivative remains unchanged

When adding or subtracting functions, how do you find the derivative?

• Add the derivatives

• Find the derivative of each term separately

• Multiply the derivatives

• Subtract the derivatives

What is the derivative of sine(x)?

• Cosine(x)

• Negative cosine(x)

• Negative sine(x)

• Sine(x)

What is the derivative of cosine(x)?

• Sine(x)

• Negative sine(x)

• Cosine(x)

• Negative cosine(x)

What is the derivative of e^x?

• e^x

• x^e

• xe^(x-1)

• e^(x-1)

How does the derivative of e^x behave?

• It is zero

• It is the same as the original function

• It is the reciprocal of the original function

• It is constant

What is the derivative of ln(x)?

• e^x

• 1/x

• ln(x)

• x

What is the derivative of a function with a logarithmic term?

• ln(x)

• 1/x

• x

• e^x

What is the first step in proving the derivative of sine using limits?

• Apply the sum identity

• Differentiate directly

• Use the limit definition

• Use the power rule