Y=(x^2+4 x)/((3 x^3+2)^4)

Name: Problem 40, Chapter 4: Calculus with Applications
Uploaded: 2019-03-31T00:18:00-08:00
Duration: 11 min 10 s
Channel: Bobby Barnes
Description: y=(x^2+4 x)/((3 x^3+2)^4)

Key Concepts

Power Rule

The power rule is a basic differentiation rule used to find the derivative of functions of the form x^n. It states that the derivative of x^n is n multiplied by x raised to the (n-1) power, and is often used in conjunction with other rules like the chain rule when differentiating more complex expressions.

Chain Rule

The chain rule is applied for differentiating composite functions, where one function is nested within another. It provides a method to differentiate a function of the form f(g(x)) by multiplying the derivative of the outer function evaluated at the inner function, f?'(g(x)), by the derivative of the inner function, g?'(x).

Differentiation

Differentiation is the process of finding the rate at which a function changes at any given point. It is a fundamental concept in calculus that involves computing the derivative of functions using various rules such as the power rule, product rule, quotient rule, and chain rule.

Quotient Rule

The quotient rule is used to differentiate functions that are expressed as one function divided by another. In general, if a function is given as f(x)/g(x), its derivative is found by taking (g(x)f?'(x) - f(x)g?'(x)) divided by [g(x)]².

Transcript

00:01 So if we want to take the derivative of the function, y is equal to x squared plus 4x divided by x3x cubed plus 2 raised to the 4th power, you might remember that we solve problems like this in the quotient rule chapter, but since we're working with chain rule right now, maybe we should rewrite this so we can use the chain rule.

00:26 So remember that if we have something in the denominator, we can rewrite it as having a negative exponent.

00:35 So i'll have x squared plus 4x times 3x cubed plus 2 raise to the negative 4th power.

00:50 And then you might notice that we have a composition of two functions where i can think of this as h of x.

01:00 So 3x cubed plus 2 and it's inside the function x to the negative 4.

01:08 So i'll rewrite this as.

01:10 So h of x is equal to 3x cubed plus 2.

01:22 So just rewriting this, rewriting y in terms of h of x.

01:26 So i get x squared plus 4x multiplied by h of x.

01:38 Raised to the negative fourth power.

01:43 So here i have two functions being multiplied together.

01:49 So i have x squared plus 4x being multiplied by h of x to the negative fourth power.

02:02 So if i want to take the derivative of this, remember i have to use the product rule.

02:11 So remember the product rule says i will take the first function, just write it down and then multiply this function by the derivative of the other one.

02:22 So, d dx, h of x raised to the negative fourth power plus.

02:36 Now, i'll write the other function, so h of x to the negative fourth power, multiplied by the derivative of x squared plus four.

03:02 So now i can go ahead and start taking these derivatives.

03:06 So i'll go ahead and write down x squared plus 4x first.

03:13 Then to take the derivative of h of x to the negative fourth, i'll have to use the generalized power rule.

03:21 So i'll move the negative 4 out front, subtract one off of it.

03:24 So i'll have negative 4 times, so h of x raised to now the negative fifth power, and we'll have to multiply by the derivative on the inside due to chain rule.

03:44 So h prime of x plus, so h of x raised to the negative fourth power, and then i'll go ahead and do this derivative right here off on the side.

04:07 So i'll end up with d dx of x squared plus four times the derivative of x...

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