# Unit Two: Differentiation: Definition and Fundamental Properties- essential knowledge

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What do the difference quotients f(a+h)−f(a)/h and f(x)−f(a)/x−a express?

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1

What do the difference quotients f(a+h)−f(a)/h and f(x)−f(a)/x−a express?

They express the average rate of change of a function over an interval

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2

How can the instantaneous rate of change of a function at x=ax = ax=a be expressed?

It can be expressed by lim⁡h→0 f(a+h)−f(a)/h or lim⁡x→a 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).

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3

How is the derivative of f defined?

The derivative of f is the function whose value at x is lim⁡h→0 f(x+h)−f(x)/h, provided this limit exists.

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4

What are the notations for the derivative of y=f(x)

Notations for the derivative include dy/dx, f′(x), and y′.

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5

In what forms can the derivative be represented?

The derivative can be represented graphically, numerically, analytically, and verbally

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6

What is the derivative of a function at a point?

The derivative of a function at a point is the slope of the line tangent to a graph of the function at that point.

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7

How can the derivative at a point be estimated?

The derivative at a point can be estimated from information given in tables or graphs

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8

How can technology be used in relation to derivatives?

Technology can be used to calculate or estimate the value of a derivative of a function at a point.

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9

What is the relationship between differentiability and continuity?

If a function is differentiable at a point, then it is continuous at that point. If a point is not in the domain of f, then it is not in the domain of f′.

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10

Can a continuous function fail to be differentiable at a point in its domain?

Yes, a continuous function may fail to be differentiable at a point in its domain.

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11

How can the derivative be calculated for functions of the form f(x) = x^r

Direct application of the definition of the derivative and specific rules can be used to calculate the derivative.

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12

How can sums, differences, and constant multiples of functions be differentiated?

Sums, differences, and constant multiples of functions can be differentiated using derivative rules.

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13

How can the power rule be used in combination with other properties?

What specific rules can be used to find the derivatives for sine, cosine, exponential, and logarithmic functions?

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14

What specific rules can be used to find the derivatives for sine, cosine, exponential, and logarithmic functions?

Specific rules for these functions include the derivatives of sinx, cos⁡x, e^x, and lnx.

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15

How can recognizing an expression for the definition of the derivative of a function whose derivative is known be useful?

It offers a strategy for determining a limit.

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16

How can derivatives of products of differentiable functions be found?

Derivatives of products of differentiable functions can be found using the product rule

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17

How can rearranging tangent, cotangent, secant, and cosecant functions using identities help in differentiation?

It allows differentiation using derivative rules.

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