Calculus Derivatives and Rules

This video tutorial focuses on finding the derivative of exponential functions, starting with the base e. It covers the basic formula for the derivative of e^x and extends to more complex examples involving different bases. The tutorial also includes a mixed review of problems and advanced problems that require the use of product and quotient rules. The video aims to provide a comprehensive understanding of differentiating exponential functions.

What is the derivative of e to the power of x?

• e^(x+1)

• x * e^x

• x^e

• e^x

How do you find the derivative of a^u where a is not e?

• a^u * u' * ln(a)

• a^u * u'

• a^u * ln(a)

• a^u * ln(e)

What is the derivative of 3^x?

• 3^x * ln(3)

• 3^x * x

• 3^x * ln(e)

• 3^x * 3

What is the derivative of e raised to the sine of x?

• e^(sin x) * sec x

• e^(sin x) * cos x

• e^(sin x) * sin x

• e^(sin x) * tan x

Which rule is used to find the derivative of x^3 * e^(4x)?

• Power Rule

• Chain Rule

• Quotient Rule

• Product Rule

What is the derivative of x^3 * e^(4x) using the product rule?

• 3x^2 * e^(4x) + 4x^3 * e^(4x)

• 3x^2 * e^(4x) - 4x^3 * e^(4x)

• 3x^2 * e^(4x) + x^3 * e^(4x)

• 3x^2 * e^(4x) + 4 * e^(4x)

What is the formula for the quotient rule?

• (g * f' - f * g') / g^2

• (f * g' - g * f') / g^2

• (f' * g - g' * f) / f^2

• (g' * f - f' * g) / f^2

How do you simplify (e^x + e^(-x))^2 - (e^x - e^(-x))^2?

• Use the product of squares

• Use the sum of squares

• Use the quotient of squares

• Use the difference of squares

What is the result of multiplying e^(-x) by e^x?

• 1

• e^(2x)

• e^x

• e^(-x)

What is the final simplified result of the expression in the quotient rule example?

• -4 / (e^x + e^(-x))^2

• 4 / (e^x - e^(-x))^2

• 4 / (e^x + e^(-x))^2

• -4 / (e^x - e^(-x))^2