Let x=t^2+t, and let y=sint (a) Find d y / d x as a function...

Let $x=t^{2}+t,$ and let $y=\sin t$ (a) Find $d y / d x$ as a function of $t$. (b) Find $\frac{d}{d t}\left(\frac{d y}{d x}\right)$ as a function of $t$ (c) Find $\frac{d}{d x}\left(\frac{d y}{d x}\right)$ as a function of $t$. Use the Chain Rule and your answer from part (b). (d) Which of the expressions in parts (b) and (c) is $d^{2} y / d x^{2} ?$

Let $x=t^{2}+t,$ and let $y=\sin t$
(a) Find $d y / d x$ as a function of $t$.
(b) Find $\frac{d}{d t}\left(\frac{d y}{d x}\right)$ as a function of $t$
(c) Find $\frac{d}{d x}\left(\frac{d y}{d x}\right)$ as a function of $t$.
Use the Chain Rule and your answer from part (b).
(d) Which of the expressions in parts (b) and (c) is $d^{2} y / d x^{2} ?$

Key Concepts

Parametric Differentiation

Parametric differentiation is the technique used to find the derivative of a function when both the dependent and independent variables are expressed in terms of a third variable, typically called the parameter. In this context, the derivative dy/dx is computed by dividing dy/dt by dx/dt, which gives the rate of change of y with respect to x along the parametric curve.

Chain Rule

The chain rule is a fundamental rule in calculus used to differentiate composite functions. It states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function multiplied by the derivative of the inner function. In problems involving parametric equations, the chain rule is applied to relate derivatives with respect to the parameter t and derivatives with respect to x.

Higher Order Derivatives in Parametric Form

Higher order derivatives in parametric form involve differentiating the first derivative of y with respect to x further with respect to x. This requires using the chain rule to differentiate dy/dx with respect to the parameter t and then dividing by the derivative dx/dt. This process allows one to compute the curvature or concavity of the parametric curve by obtaining the second derivative d²y/dx².

Transcript

00:01 In this question, we will be talking about chain rule.

00:05 The question gives us parametric equations representing a curve in two -dimensional space, and asks us to find some derivatives related to these equations.

00:20 That is, first we want to find the derivative of y with respect to x as a function of t, then the derivative of the derivative of y with respect to x with respect to t, and then the derivative of that derivative with respect to x.

00:43 So if you didn't get that in my word explanation, my verbal explanation, this derivative gets differentiated with respect to t and with respect to x in parts b and c respectively.

01:00 After that, we're also going to answer a question for part d.

01:05 So first of all, we have d -y -d -x, and we want to use chain rule, as well as the these parametric equations to find it.

01:17 Now, chain rule tells us that if we multiply this by dx d t, we get d y to t.

01:30 Because we have x and y in terms of t, these are more relevant derivatives, because we can differentiate y with respect to t directly to find this, and x with respect to directly to find this.

01:49 We can't be so direct about d, y, d, x, because we don't have an expression of y in terms of x.

02:01 And so, uh, let's find these and then take their ratio to find dydx.

02:17 Now d .yd .t is the derivative of sine t with respect to t.

02:23 Dx, dt is the derivative of t squared plus t with respect to t.

02:31 And so this is the derivative of y with respect to t.

02:35 It's pretty simple.

02:44 For part b now, we want to differentiate this expression in terms of t, this derivative, with respect to t...