Differentiate the function by forming the difference...

Key Concepts

Binomial Theorem

The Binomial Theorem is used to expand expressions of the form (x+h)^n into a sum involving terms of x and h. This expansion is essential when applying the difference quotient to polynomial functions, as it allows for the cancellation of terms when taking the limit.

Limit Process

The limit process in differentiation involves taking the limit of the difference quotient as the change in the input approaches zero. This step is crucial because it transitions from an average rate of change to the instantaneous rate of change, leading to the exact derivative.

Algebraic Simplification

Algebraic simplification refers to the process of reducing complex expressions to simpler ones by combining like terms and canceling common factors. In differentiation using the difference quotient, simplification is key to eliminating the variable h from the denominator so that the limit can be properly evaluated.

Differentiation

Differentiation is the process in calculus by which we determine the rate at which one quantity changes with respect to another. It involves computing the derivative, which provides important information such as slopes of curves and instantaneous rates of change.

Difference Quotient

The difference quotient is a formula used to compute the derivative of a function by evaluating the average rate of change over an interval and then taking the limit as the interval shrinks to zero. It is defined as [f(x+h) - f(x)]/h and serves as the foundation for the limit definition of the derivative.

Transcript

00:01 X to the fourth we want to find the derivative f prime of x by finding the limit of this difference quotient as h goes towards zero so we need to take f of x plus h well f of x is x to the fourth so f of x plus h will be x plus h to the fourth then we have to subtract f of x which is x to the fourth then we have to put that over h.

00:47 And we have to take the limit of this.

00:50 Let me squeeze it in here.

00:51 I forgot to write it in.

00:52 We have to take the limit of this quotient as h goes towards zero.

00:59 So we need to take the limit of x plus h to the fourth minus x to the fourth, divided by h as h goes towards zero.

01:07 So we're going to expand l x plus h to the fourth.

01:16 Okay, so x plus h to the fourth expands out to be this entire expression from x to the fourth all the way down to plus h to the fourth.

01:28 And then we still have to subtract x to the fourth.

01:31 So we wrote it here and we have to put that all over h.

01:34 And we need to take the limit of this entire expression as h goes towards zero.

01:45 Now we could do a little bit of canceling, not a whole lot.

01:49 X to the fourth minus x to the fourth cancels.

01:54 So we are going to take the limit as h approaches zero of this entire expression over h.

02:06 Now, i'm going to factor out of these terms in the numerator, the greatest common factor, which would be just h.

02:18 So out of the numerator, we're going to factor out in h.

02:22 So h times 4x cubed.

02:29 That would give you to 4x cubed h plus...

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