Exploring Derivatives of Polynomials and Exponential Functions

This video tutorial covers the basics of derivatives, focusing on the power rule and its application to polynomial and exponential functions. It explains the sum and difference rules for derivatives and demonstrates how to find horizontal tangent lines. The tutorial also explores the unique properties of exponential functions, emphasizing the importance of not applying the power rule to them. Through examples, the video illustrates these concepts, providing a comprehensive understanding of derivatives in calculus.

What is the power rule for finding the derivative of a function f(x) = x^n?

• n * x^(n-1)

• n * x^(n+1)

• x^n

• n * x^n

If y = x^{1000}, what is the derivative y'?

• x^{999}

• 999 * x^{1000}

• 1000 * x^{1000}

• 1000 * x^{999}

What is the derivative of y = t^4 with respect to t?

• t^3

• 4 * t^3

• 3 * t^4

• 4 * t^4

According to the sum rule, what is the derivative of f(x) + g(x) if both functions are differentiable?

• f'(x) + g'(x)

• f(x) + g(x)

• f(x) * g(x)

• f'(x) * g'(x)

What is the derivative of a constant function?

• 0

• 1

• Undefined

• The constant itself

For the function y = x^4 - 6x^2 + 4, where is the tangent line horizontal?

• x = 0 or x = ±3

• x = ±3

• x = ±√3

• x = 0 or x = ±√3

What is the derivative of an exponential function f(x) = a^x using limit laws?

• a^x

• a^(x-1)

• a^x * ln(a)

• x * a^(x-1)

What is the special property of the natural exponential function e^x?

• Its derivative is itself

• Its derivative is zero

• Its derivative is x

• Its derivative is e

What should you never apply to e^x when finding its derivative?

• The product rule

• The sum rule

• The power rule

• The quotient rule

If f(x) = e^x - x, what is the second derivative f''(x)?

• 1

• e^x - 1

• 0

• e^x