Find the derivative of the function using the definition of derivative. State the domain of the

Find the derivative of the function using the definition of...

Key Concepts

Definition of Derivative

The derivative of a function is defined as the limit of the difference quotient, (f(x + h) - f(x))/h, as h approaches zero. This concept represents the instantaneous rate of change of the function at any given point and is foundational in calculus.

Difference Quotient

The difference quotient is the expression (f(x + h) - f(x))/h that forms the basis for the derivative when its limit as h approaches zero is taken. It quantifies the average rate of change over an interval, which converges to the instantaneous rate of change in the limit.

Polynomial Functions

Polynomial functions are algebraic expressions involving sums of powers of x with constant coefficients. They are known for being smooth, continuous, and differentiable everywhere on the real line, which means that both the function and its derivative have domains that include all real numbers.

Domain of a Function

The domain of a function is the complete set of all possible input values (x-values) for which the function is defined. For polynomial functions, the domain is typically all real numbers because there are no restrictions like division by zero or square roots of negative numbers.

Limit Concept

The concept of limits is crucial in defining the derivative and understanding continuity. It involves determining the value that a function approaches as its input gets arbitrarily close to a given point, which is essential in evaluating the derivative using the definition.

Transcript

00:01 Okay, here we have the function f of x equals 4 plus 8x minus 5x squared.

00:08 The domain of a function are all the values that can be inputted into the function.

00:16 So basically all values of x for which this function would be defined.

00:20 This function is going to be defined for every real value of x.

00:24 So we're going to say the domain is the entire x -axis or all real numbers.

00:32 Now we have to find the derivative of this function using the definition of the derivative.

00:44 So f prime of x is going to be the limit as h approaches zero of f evaluated at x plus h minus f evaluated at x, all divided by h as h approaches 0.

01:33 So define f prime of x, we have to take the limit of this function evaluated at x plus h minus the value of the function at x given right here, all divided by h as h approaches zero.

01:48 So what we're going to do first is we're going to substitute x plus h everywhere in you see x.

01:53 Then we're going to subtract the function f of x written here, divide it all by h, do a little bit of algebra, and take the limit as h approaches zero.

02:09 So f, evaluated x plus h will be 4 plus 8 times x plus h minus 5 times x plus h squared.

02:20 And then we have to subtract f of x.

02:23 So we're going to subtract each of these terms in f of x.

02:26 So we're going to subtract the 4, subtract the 8x, and then we're also going to subtract the minus 5x squared, which turns into a plus 5x squared.

02:35 All of that is divided by h, and we're taking the limit as h approaches zero.

02:42 Continuing simplifying this with a little bit of algebra, this is going to be multiplying the x plus h, so we'll distribute it 8x plus 8h...

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