Differentiation: Definition and Fundamental Properties · ⁨الاشتقاق: التعريف والخصائص الأساسية⁩

Enduring Understanding (CHA-2): Derivatives allow us to determine rates of change at an instant by applying limits to knowledge about rates of change over intervals.

Learning Objective CHA-2.A: Determine average rates of change using difference quotients.

• CHA-2.A.1 The difference quotients $\dfrac{f(a+h)-f(a)}{h}$ and $\dfrac{f(x)-f(a)}{x-a}$ express the average rate of change of a function over an interval.

Learning Objective CHA-2.B: Represent the derivative of a function as the limit of a difference quotient.

• CHA-2.B.1 The instantaneous rate of change of a function at $x=a$ can be expressed by $\lim\limits_{h\to 0}\dfrac{f(a+h)-f(a)}{h}$ or $\lim\limits_{x\to a}\dfrac{f(x)-f(a)}{x-a}$, provided the limit exists. These are equivalent forms of the definition of the derivative and are denoted $f'(a)$.

Enduring Understanding (CHA-2): Derivatives allow us to determine rates of change at an instant by applying limits to knowledge about rates of change over intervals.

Learning Objective CHA-2.B: Represent the derivative of a function as the limit of a difference quotient.

• CHA-2.B.2 The derivative of $f$ is the function whose value at $x$ is $\lim\limits_{h\to 0}\dfrac{f(x+h)-f(x)}{h}$, provided this limit exists.
• CHA-2.B.3 For $y=f(x)$, notations for the derivative include $\dfrac{dy}{dx}$, $f'(x)$, and $y'$.
• CHA-2.B.4 The derivative can be represented graphically, numerically, analytically, and verbally.

Learning Objective CHA-2.C: Determine the equation of a line tangent to a curve at a given point.

• CHA-2.C.1 The derivative of a function at a point is the slope of the line tangent to a graph of the function at that point.

Enduring Understanding (CHA-2): Derivatives allow us to determine rates of change at an instant by applying limits to knowledge about rates of change over intervals.

Learning Objective CHA-2.D: Estimate derivatives.

• CHA-2.D.1 The derivative at a point can be estimated from information given in tables or graphs.
• CHA-2.D.2 Technology can be used to calculate or estimate the value of a derivative of a function at a point.

Enduring Understanding (FUN-2): Recognizing that a function's derivative may also be a function allows us to develop knowledge about the related behaviors of both.

Learning Objective FUN-2.A: Explain the relationship between differentiability and continuity.

• FUN-2.A.1 If a function is differentiable at a point, then it is continuous at that point. In particular, if a point is not in the domain of $f$, then it is not in the domain of $f'$.
• FUN-2.A.2 A continuous function may fail to be differentiable at a point in its domain.
  • Illustrative examples for FUN-2.A.2:
    • The left hand and right hand limits of the difference quotient are not equal, as in $f(x)=|x|$ at $x=0$.
    • The tangent line is vertical and has no slope, as in $f(x)=\sqrt[3]{x}$ at $x=0$.

Enduring Understanding (FUN-3): Recognizing opportunities to apply derivative rules can simplify differentiation.

Learning Objective FUN-3.A: Calculate derivatives of familiar functions.

• FUN-3.A.1 Direct application of the definition of the derivative and specific rules can be used to calculate the derivative for functions of the form $f(x)=x^{r}$.

Enduring Understanding (FUN-3): Recognizing opportunities to apply derivative rules can simplify differentiation.

Learning Objective FUN-3.A: Calculate derivatives of familiar functions.

• FUN-3.A.2 Sums, differences, and constant multiples of functions can be differentiated using derivative rules.
• FUN-3.A.3 The power rule combined with sum, difference, and constant multiple properties can be used to find the derivatives for polynomial functions.

Enduring Understanding (FUN-3): Recognizing opportunities to apply derivative rules can simplify differentiation.

Learning Objective FUN-3.A: Calculate derivatives of familiar functions.

• FUN-3.A.4 Specific rules can be used to find the derivatives for sine, cosine, exponential, and logarithmic functions.

Enduring Understanding (LIM-3): Reasoning with definitions, theorems, and properties can be used to determine a limit.

Learning Objective LIM-3.A: Interpret a limit as a definition of a derivative.

• LIM-3.A.1 In some cases, recognizing an expression for the definition of the derivative of a function whose derivative is known offers a strategy for determining a limit.

Enduring Understanding (FUN-3): Recognizing opportunities to apply derivative rules can simplify differentiation.

Learning Objective FUN-3.B: Calculate derivatives of products and quotients of differentiable functions.

• FUN-3.B.1 Derivatives of products of differentiable functions can be found using the product rule.

Enduring Understanding (FUN-3): Recognizing opportunities to apply derivative rules can simplify differentiation.

Learning Objective FUN-3.B: Calculate derivatives of products and quotients of differentiable functions.

• FUN-3.B.2 Derivatives of quotients of differentiable functions can be found using the quotient rule.

Enduring Understanding (FUN-3): Recognizing opportunities to apply derivative rules can simplify differentiation.

Learning Objective FUN-3.B: Calculate derivatives of products and quotients of differentiable functions.

• FUN-3.B.3 Rearranging tangent, cotangent, secant, and cosecant functions using identities allows differentiation using derivative rules.