Topic 1: Definition of a Derivative and Its Geometric Interpretation
Easy Questions 🌱
Find the derivative of \( f(x) = x^2 \) using the limit definition.
Compute the slope of \( f(x) = 3x + 2 \) at \( x = 1 \) using the definition.
Determine the derivative of \( f(x) = x^3 \) at \( x = 0 \) using first principles.
Moderate Questions 🌱
Find the derivative of \( f(x) = \sqrt{x} \) at \( x = 4 \) using the limit definition.
Compute the slope of \( f(x) = \frac{1}{x} \) at \( x = 2 \) using the definition.
Determine the derivative of \( f(x) = x^2 + 2x \) at \( x = -1 \) using first principles.
Hard Questions 🌟
Find the derivative of \( f(x) = \sin x \) at \( x = 0 \) using the limit definition.
Compute the derivative of \( f(x) = e^x \) at \( x = 1 \) using first principles.
Topic 2: Basic Differentiation Rules
Easy Questions 🌱
Find the derivative of \( f(x) = 5x^3 \).
Compute \( \frac{d}{dx} (2x^2 + 3) \).
Determine the derivative of \( f(x) = 4x - 7 \).
Moderate Questions 🌱
Find the derivative of \( f(x) = 3x^4 - 2x^2 + 1 \).
Compute \( \frac{d}{dx} (x^5 - 4x + 2) \).
Determine the derivative of \( f(x) = 6x^{-2} + 3x \).
Hard Questions 🌟
Find the derivative of \( f(x) = \frac{1}{x^3} + x^{-4} \).
Compute \( \frac{d}{dx} (x^{1/3} - x^{-2/3}) \).
Topic 3: Product and Quotient Rules
Easy Questions 🌱
Find the derivative of \( f(x) = x^2 \cdot e^x \) using the product rule.
Compute \( \frac{d}{dx} \left( \frac{x}{x + 1} \right) \) using the quotient rule.
Determine the derivative of \( f(x) = x \cdot \sin x \).
Moderate Questions 🌱
Find the derivative of \( f(x) = x^3 e^{-x} \) using the product rule.
Compute \( \frac{d}{dx} \left( \frac{1}{x^2 + 1} \right) \) using the quotient rule.
Determine the derivative of \( f(x) = \cos x \cdot x^2 \).
Hard Questions 🌟
Find the derivative of \( f(x) = x^2 e^{x^2} \) using the product rule.
Compute \( \frac{d}{dx} \left( \frac{e^x}{x^2 + 1} \right) \) using the quotient rule.
Topic 4: Chain Rule and Implicit Differentiation
Easy Questions 🌱
Find the derivative of \( f(x) = (x^2 + 1)^3 \) using the chain rule.
Compute \( \frac{d}{dx} (\sin(x^2)) \) using the chain rule.
Determine \( \frac{dy}{dx} \) if \( x^2 + y^2 = 1 \) using implicit differentiation.
Moderate Questions 🌱
Find the derivative of \( f(x) = e^{x^3} \) using the chain rule.
Compute \( \frac{d}{dx} (\ln(x^2 + 1)) \) using the chain rule.
Determine \( \frac{dy}{dx} \) if \( x^3 + y^3 = 6xy \) using implicit differentiation.
Hard Questions 🌟
Find the derivative of \( f(x) = \sin^3(x^2) \) using the chain rule.
Compute \( \frac{dy}{dx} \) if \( \sin(x + y) = x^2 \) using implicit differentiation.
Topic 5: Higher-Order Derivatives
Easy Questions 🌱
Find the second derivative of \( f(x) = x^3 \).
Compute the first derivative of \( f(x) = 2x^2 + 3x \).
Moderate Questions 🌱
Find the third derivative of \( f(x) = x^4 \).
Compute the second derivative of \( f(x) = \sin x \).
Hard Questions 🌟
Find the fourth derivative of \( f(x) = e^x \).
Compute the third derivative of \( f(x) = x^2 \cos x \).
Topic 6: Applications of Derivatives (Tangent Lines)
Easy Questions 🌱
Find the equation of the tangent line to \( f(x) = x^2 \) at \( x = 1 \).
Compute the slope of the tangent to \( f(x) = 2x + 3 \) at \( x = 0 \).
Moderate Questions 🌱
Find the tangent line to \( f(x) = x^3 \) at \( x = -1 \).
Compute the equation of the tangent to \( f(x) = \sin x \) at \( x = \pi/2 \).
Hard Questions 🌟
Find the tangent line to \( f(x) = e^x \) at \( x = 1 \).
Compute the tangent line to \( f(x) = x^2 \ln x \) at \( x = 1 \).
Topic 7: Applications of Derivatives (Motion)
Easy Questions 🌱
Find the velocity of \( s(t) = t^2 \) at \( t = 2 \).
Compute the acceleration of \( v(t) = 3t \) at \( t = 1 \).
Moderate Questions 🌱
Find the velocity of \( s(t) = t^3 - 2t \) at \( t = 1 \).
Compute the acceleration of \( v(t) = \sin t \) at \( t = \pi/2 \).
Hard Questions 🌟
Find the velocity and acceleration of \( s(t) = e^t \sin t \) at \( t = 0 \).
Compute the acceleration of \( v(t) = t^2 e^{-t} \) at \( t = 1 \).
Topic 8: Applications of Derivatives (Related Rates)
Easy Questions 🌱
If \( A = x^2 \) and \( x \) increases at 2 units/sec, find \( \frac{dA}{dt} \) when \( x = 3 \).
Find \( \frac{dy}{dt} \) if \( y = x^3 \) and \( x = 2 \) when \( \frac{dx}{dt} = 1 \).
Moderate Questions 🌱
If \( V = \frac{4}{3}\pi r^3 \) and \( r \) increases at 2 cm/sec, find \( \frac{dV}{dt} \) when \( r = 3 \).
Compute \( \frac{dA}{dt} \) if \( A = \pi r^2 \) and \( r = 5 \) when \( \frac{dr}{dt} = 0.1 \).
Hard Questions 🌟
If \( x^2 + y^2 = 25 \) and \( x \) decreases at 1 unit/sec, find \( \frac{dy}{dt} \) when \( x = 3 \), \( y = 4 \).
Compute \( \frac{dV}{dt} \) if \( V = \frac{1}{3}\pi r^2 h \) and \( r \) and \( h \) change with \( \frac{dr}{dt} = 2 \), \( \frac{dh}{dt} = 3 \) when \( r = 1 \), \( h = 2 \).
Topic 9: Increasing/Decreasing Functions and Extrema
Easy Questions 🌱
Determine where \( f(x) = x^2 \) is increasing.
Find the critical points of \( f(x) = x^3 - 3x \).
Moderate Questions 🌱
Determine where \( f(x) = x^3 - 6x \) is decreasing.
Find the local maxima of \( f(x) = x^4 - 2x^2 \).
Hard Questions 🌟
Determine the intervals where \( f(x) = x^2 e^{-x} \) is increasing.
Find all extrema of \( f(x) = \sin x + \cos x \) on \( [0, 2\pi] \).
Topic 10: Concavity, Inflection Points, and the Second Derivative Test
Easy Questions 🌱
Find where \( f(x) = x^2 \) is concave up.
Compute the second derivative of \( f(x) = x^3 \).
Moderate Questions 🌱
Determine the inflection points of \( f(x) = x^3 - 3x \).
Find where \( f(x) = x^4 \) is concave down.
Hard Questions 🌟
Determine the concavity and inflection points of \( f(x) = x e^{-x} \).
Use the second derivative test to classify critical points of \( f(x) = x^3 - 3x + 1 \).
Topic 11: Curve Sketching Using Derivatives
Easy Questions 🌱
Sketch \( f(x) = x^2 \) using its derivative.
Find the critical points of \( f(x) = x^3 - 3x \).
Moderate Questions 🌱
Sketch \( f(x) = x^3 - 6x \) using first and second derivatives.
Determine the behavior of \( f(x) = x^4 - 2x^2 \) using derivatives.
Hard Questions 🌟
Sketch \( f(x) = x^2 e^{-x} \) using all derivative information.
Analyze \( f(x) = \sin x + \cos x \) on \( [0, 2\pi] \) with derivatives.
Topic 12: L'Hôpital’s Rule and Indeterminate Forms
Easy Questions 🌱
Evaluate \( \lim_{x \to 0} \frac{\sin x}{x} \) using L'Hôpital’s rule.
Find \( \lim_{x \to \infty} \frac{2x}{x + 1} \).
Moderate Questions 🌱
Compute \( \lim_{x \to 0} \frac{e^x - 1}{x} \) using L'Hôpital’s rule.
Evaluate \( \lim_{x \to 1} \frac{x^2 - 1}{x - 1} \) with L'Hôpital’s rule.
Hard Questions 🌟
Find \( \lim_{x \to 0^+} \frac{\ln x}{1/x} \) using L'Hôpital’s rule.
Evaluate \( \lim_{x \to 0} \frac{\sin x - x}{x^3} \) with L'Hôpital’s rule.