Find (d y)/(d x) at x=1 if y=(t^2+2)/(t^2-2) and t=x^3 |...

Name: Problem 17, Chapter :
Uploaded: 2019-09-24T20:02:41-08:00
Duration: 2 min 23 s
Channel: Taylor Shimono
Description: Find (d y)/(d x) at x=1 if y=(t^2+2)/(t^2-2) and t=x^3

Key Concepts

Composite Function Differentiation

Differentiating composite functions involves applying the chain rule to handle functions that are nested within one another. When a function is written as one function of another, this process allows us to systematically break down the computation of the derivative by differentiating the outer function with respect to the inner variable and then multiplying by the derivative of the inner function.

Power Rule

The power rule is one of the most basic rules for differentiation, providing a quick method to differentiate functions of the form x^n. It states that the derivative of x raised to a constant exponent n is n times x raised to the power of (n-1), making it a critical tool for handling polynomial functions in calculus.

Chain Rule

The chain rule is a fundamental technique in calculus used to differentiate composite functions, that is, functions formed by the composition of two or more functions. It states that the derivative of a composite function is the product of the derivative of the outer function evaluated at the inner function and the derivative of the inner function.

Quotient Rule

The quotient rule is a method for finding the derivative of a function that is expressed as a ratio of two differentiable functions. According to this rule, if a function is expressed as a numerator function divided by a denominator function, its derivative is obtained by taking the derivative of the numerator times the denominator minus the numerator times the derivative of the denominator, all over the square of the denominator.

Transcript

00:01 So for this problem, we're solving for a dydx at x is equal to 1.

00:05 But we can see that when we look at our equation for y, we have y in terms of t.

00:09 And we also have an equation for t in terms of x.

00:12 But if we think about what happens when we take the derivative, we know that for dy, we would take the derivative of dy with respect to t.

00:19 And if we take the derivative of t with respect to x, we're going to get dtdx.

00:24 So if we multiply these two terms together, we're going to be able to cancel out these dts and just be left with dydx.

00:31 So essentially we just need to find the derivative of each of these individual functions.

00:35 So we can start by finding dydt, and we can see that we're going to need to use the quotient rule here.

00:42 So we start by finding the derivative of the numerator, which in this case is just 2t, times the denominator unchanged, which will be times t squared minus 2.

00:51 And we're going to subtract the numerator unchanged, so t squared plus 2, times the derivative of our denominator, which is just 2t.

00:58 And this is divided by our denominator squared, which will be t squared minus 2 squared.

01:05 And we can go ahead and simplify this.

01:07 And we can see that when we factor this, or excuse me, distribute this out, we're going to essentially just be left with our numerator of negative 8t, and this is over our denominator of t squared minus 2 squared...

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