Differentiation
Differentiation
Differentiation
what you'll learn
Lesson details
Unit 3: Foundations of Differentiation and First Derivatives
3.1 Concept of the Derivative
Limit Definition: f′(x)=limh→0f(x+h)−f(x)hf'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}f′(x)=limh→0hf(x+h)−f(x), geometric interpretation.
Tangent and Secant Lines: Visual understanding of instantaneous vs. average rate of change.
Notation: f′(x),dydx,Df(x)f'(x), \frac{dy}{dx}, Df(x)f′(x),dxdy,Df(x), interchangeable use and context.
3.2 Rules of Differentiation
Power Rule: ddx(xn)=nxn−1\frac{d}{dx}(x^n) = nx^{n-1}dxd(xn)=nxn−1.
Constant & Constant Multiple Rules: ddx(c)=0\frac{d}{dx}(c) = 0dxd(c)=0, ddx(cf(x))=cf′(x)\frac{d}{dx}(cf(x)) = c f'(x)dxd(cf(x))=cf′(x).
Sum and Difference Rules: Differentiating term by term.
3.3 Applications of First Derivatives
Gradient Function: Relationship between function and its derivative.
Turning Points: Local maxima and minima via f′(x)=0f'(x) = 0f′(x)=0.
Increasing/Decreasing Intervals: Sign of f′(x)f'(x)f′(x) determines function behavior.
Sketching Derivative Graphs: From a given f(x)f(x)f(x), or matching derivative to original graph.
3.4 Real-World Applications
Rates of Change: Velocity from displacement, marginal cost from total cost.
Optimisation Problems: Finding best or worst values under constraints.
Motion Analysis: Interpreting velocity and acceleration using s(t),v(t),a(t)s(t), v(t), a(t)s(t),v(t),a(t).
Unit 4: Advanced Differentiation and Second Derivatives
4.1 Exponential and Trigonometric Differentiation
Exponential Rule
Trigonometric Rules
Product and Quotient Rules: Handling products and divisions of functions.
Chain Rule: Differentiating composite functions
4.2 Second Derivative and Curve Concavity
Second Derivative
Concavity & Inflection Points
Sketching with Second Derivatives: Visual features of inflection, curvature.
4.3 Problem Solving and Modelling Contexts
Applied Problems: Maximising volume/surface area, minimising cost/time.
Interpreting Graphs: Translating real scenarios into differentiation-based models.
Multiple Representations: Function table, graph, and derivative all tied together.
4.4 Technology Integration
Calculator Derivative Tools: Numerical and symbolic differentiation.
Graphing Derivatives: Using TI-Nspire/Casio to verify slopes, turning points.
Instantaneous Rate Approximations: From plotted data or discrete points.
Unit 3: Foundations of Differentiation and First Derivatives
3.1 Concept of the Derivative
Limit Definition: f′(x)=limh→0f(x+h)−f(x)hf'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}f′(x)=limh→0hf(x+h)−f(x), geometric interpretation.
Tangent and Secant Lines: Visual understanding of instantaneous vs. average rate of change.
Notation: f′(x),dydx,Df(x)f'(x), \frac{dy}{dx}, Df(x)f′(x),dxdy,Df(x), interchangeable use and context.
3.2 Rules of Differentiation
Power Rule: ddx(xn)=nxn−1\frac{d}{dx}(x^n) = nx^{n-1}dxd(xn)=nxn−1.
Constant & Constant Multiple Rules: ddx(c)=0\frac{d}{dx}(c) = 0dxd(c)=0, ddx(cf(x))=cf′(x)\frac{d}{dx}(cf(x)) = c f'(x)dxd(cf(x))=cf′(x).
Sum and Difference Rules: Differentiating term by term.
3.3 Applications of First Derivatives
Gradient Function: Relationship between function and its derivative.
Turning Points: Local maxima and minima via f′(x)=0f'(x) = 0f′(x)=0.
Increasing/Decreasing Intervals: Sign of f′(x)f'(x)f′(x) determines function behavior.
Sketching Derivative Graphs: From a given f(x)f(x)f(x), or matching derivative to original graph.
3.4 Real-World Applications
Rates of Change: Velocity from displacement, marginal cost from total cost.
Optimisation Problems: Finding best or worst values under constraints.
Motion Analysis: Interpreting velocity and acceleration using s(t),v(t),a(t)s(t), v(t), a(t)s(t),v(t),a(t).
Unit 4: Advanced Differentiation and Second Derivatives
4.1 Exponential and Trigonometric Differentiation
Exponential Rule
Trigonometric Rules
Product and Quotient Rules: Handling products and divisions of functions.
Chain Rule: Differentiating composite functions
4.2 Second Derivative and Curve Concavity
Second Derivative
Concavity & Inflection Points
Sketching with Second Derivatives: Visual features of inflection, curvature.
4.3 Problem Solving and Modelling Contexts
Applied Problems: Maximising volume/surface area, minimising cost/time.
Interpreting Graphs: Translating real scenarios into differentiation-based models.
Multiple Representations: Function table, graph, and derivative all tied together.
4.4 Technology Integration
Calculator Derivative Tools: Numerical and symbolic differentiation.
Graphing Derivatives: Using TI-Nspire/Casio to verify slopes, turning points.
Instantaneous Rate Approximations: From plotted data or discrete points.
Unit 3: Foundations of Differentiation and First Derivatives
3.1 Concept of the Derivative
Limit Definition: f′(x)=limh→0f(x+h)−f(x)hf'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}f′(x)=limh→0hf(x+h)−f(x), geometric interpretation.
Tangent and Secant Lines: Visual understanding of instantaneous vs. average rate of change.
Notation: f′(x),dydx,Df(x)f'(x), \frac{dy}{dx}, Df(x)f′(x),dxdy,Df(x), interchangeable use and context.
3.2 Rules of Differentiation
Power Rule: ddx(xn)=nxn−1\frac{d}{dx}(x^n) = nx^{n-1}dxd(xn)=nxn−1.
Constant & Constant Multiple Rules: ddx(c)=0\frac{d}{dx}(c) = 0dxd(c)=0, ddx(cf(x))=cf′(x)\frac{d}{dx}(cf(x)) = c f'(x)dxd(cf(x))=cf′(x).
Sum and Difference Rules: Differentiating term by term.
3.3 Applications of First Derivatives
Gradient Function: Relationship between function and its derivative.
Turning Points: Local maxima and minima via f′(x)=0f'(x) = 0f′(x)=0.
Increasing/Decreasing Intervals: Sign of f′(x)f'(x)f′(x) determines function behavior.
Sketching Derivative Graphs: From a given f(x)f(x)f(x), or matching derivative to original graph.
3.4 Real-World Applications
Rates of Change: Velocity from displacement, marginal cost from total cost.
Optimisation Problems: Finding best or worst values under constraints.
Motion Analysis: Interpreting velocity and acceleration using s(t),v(t),a(t)s(t), v(t), a(t)s(t),v(t),a(t).
Unit 4: Advanced Differentiation and Second Derivatives
4.1 Exponential and Trigonometric Differentiation
Exponential Rule
Trigonometric Rules
Product and Quotient Rules: Handling products and divisions of functions.
Chain Rule: Differentiating composite functions
4.2 Second Derivative and Curve Concavity
Second Derivative
Concavity & Inflection Points
Sketching with Second Derivatives: Visual features of inflection, curvature.
4.3 Problem Solving and Modelling Contexts
Applied Problems: Maximising volume/surface area, minimising cost/time.
Interpreting Graphs: Translating real scenarios into differentiation-based models.
Multiple Representations: Function table, graph, and derivative all tied together.
4.4 Technology Integration
Calculator Derivative Tools: Numerical and symbolic differentiation.
Graphing Derivatives: Using TI-Nspire/Casio to verify slopes, turning points.
Instantaneous Rate Approximations: From plotted data or discrete points.
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