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

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

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).

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.

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

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

In what forms can the derivative be represented?

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

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.

How can the derivative at a point be estimated?

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

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.

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′.

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.

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.

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

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

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?

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.

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.

How can derivatives of products of differentiable functions be found?

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

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

It allows differentiation using derivative rules.