## What is the chain rule?

The chain rule is a method used to find the derivative of a function that is composed of multiple other functions. In other words, it allows you to find the derivative of a ” chained ” function together with other functions.

For example, suppose we have a function f(x) that is equal to g(h(x)). We can use the chain rule to find the derivative of this function by first finding the derivatives of g and h, and then combining them using the chain rule.

The chain rule can be written in mathematical notation as follows:

(f(g(x)))’ = f'(g(x)) * g'(x)

As you can see, the chain rule states that the derivative of a function that is composed of multiple other functions is equal to the derivative of theoutermost function multiplied by the derivative ofthe innermost function.

## How to use the chain rule to find derivatives

The chain rule is a way to find derivatives of composite functions. A composite function is a function that is made up of two or more functions. The chain rule is a way to find derivatives of composite functions. In this section, we’ll show you how to use the chain rule to find derivatives.

### Find the derivative of z with respect to t

We can use the chain rule to find the derivative of z with respect to t. First, we need to find the derivative of z with respect to x and y. We can do this by taking the partial derivative of each term in the equation with respect to x and y.

\frac{\partial z}{\partial x} = 2x \ y^2 + y \ xy + sint \ x

\frac{\partial z}{\partial y} = 2y \ x^2 + x \ xy – 9et \ y

Now that we have the derivatives of z with respect to x and y, we can use the chain rule to find the derivative of z with respect to t.

\frac{dz}{dt} = \frac{\partial z}{\partial x}\frac{dx}{dt} + \frac{\partial z}{\partial y}\frac{dy}{dt}

plugging in our values for the derivatives of z with respect to x and y, we get:

\frac{dz}{dt} = (2x \ y^2 + y \ xy + sint \ x) (2t) + (2y \ x^2 + x \ xy – 9et \ y) (9t)

= 4xty^2 + 2xy^2 + 2sintx+ 18tyx^2 – 81etxy

### Find the derivative of x^2 with respect to t

To find the derivative of x2 with respect to t, we use the chain rule. The chain rule states that if we have a function that is a composition of two functions, we can take the derivative of the outer function with respect to the inner function, and then multiply that by the derivative of the inner function with respect to the variable we’re interested in. In this case, we have a function z that is equal to x2 multiplied by y2. We can take the derivative of z with respect to x and multiply that by the derivative of x with respect to t. This gives us:

dzdx * dxdt = 2x * 1 = 2xt

So, the derivative of x2 with respect to t is 2xt.

### Find the derivative of y^2 with respect to t

To find the derivative of y^2 with respect to t, we can use the chain rule. First, we need to find the derivative of z with respect to t. Since z = x^2 + y^2, we can take the derivative of each term separately. For the first term, we have dz/dx = 2x, and for the second term we have dz/dy = 2y. Now, we need to find the derivative of x with respect to t. We can see that x = st, so dx/dt = s. For y, we have dy/dt = 9e^t. Plugging these derivatives into the chain rule formula, we get:

dz/dt = (dz/dx)(dx/dt) + (dz/dy)(dy/dt)

= (2x)(s) + (2y)(9e^t)

= 2xs + 18ye^t

### Find the derivative of xy with respect to t

The chain rule states that if you have a function of the form f(g(x)), then the derivative of that function is equal to f'(g(x)) * g'(x).

To find the derivative of xy with respect to t, first we need to find the derivative of both x and y with respect to t. We can do this using the chain rule:

dxdt = 2x * cos(t)

dydt = 9e^t

### Find the derivative of x with respect to t

We can use the chain rule to find the derivative of x with respect to t. First, we need to find the derivative of z with respect to t. We are given that z = x^2 + y^2 + xy +x*sin(t) + y*9*e^t. Taking the derivative of each term with respect to t, we get:

dz/dt = 2x*dx/dt + 2y*dy/dt + x*dy/dt + y*9*e^t + sin(t)*dx/dt

Now we can plug in the values for dx/dt and dy/dt. We are given that x = sint and y = 9et. So:

dz/dt = 2x*cos(t) + 2y*9e^t + x*9e^t + y*9e^t + sin(t)*cos(t)

dz/dt = 2sint*cos(t) + 2(9et)*9e^t + sint*9e^t+(9et)*9e^t+sint*cos(t)

dz/dt = 18sint*cos(t)+18et+27et+sint*cos(t)

dz/dt= 45sintcost+18e ˆ2 ˆ t 3 27 e 1 cos t

### Find the derivative of sin(t) with respect to t

To find the derivative of sin(t) with respect to t, we can use the chain rule.

The chain rule states that if we have a function z that is a function of x which is itself a function of t, then we can find the derivative of z with respect to t by first finding the derivative of z with respect to x and then multiplying by the derivative of x with respect to t.

In our case, z is equal to x^2 * y^2 * x * y * sin(t). We can take the derivative of each term in this expression separately and then multiply them all together to get the final answer.

The derivative of x^2 with respect to x is 2x, so the derivative of x^2 with respect to t is 2x * (dy/dt). The derivative of y^2 with respect to y is 2y, so the derivative of y^2 with respect to t is 2y * (dx/dt). The derivative of sin(t) with respect to t is cos(t), so the final answer is 2x * (dy/dt) + 2y * (dx/dt) * cos(t).

## How to use the chain rule to find integrals

The chain rule is one of the most important rules in calculus, and it can be used to find derivatives of composite functions. In this section, we’ll go over how to use the chain rule to find integrals. We’ll also talk about when and how to use the chain rule.

### Find the integral of z with respect to t

To find the integral of z with respect to t, we will use the chain rule. The chain rule states that if z is a function of x and y, and x and y are both functions of t, then we can find the derivative of z with respect to t by first finding the derivative of z with respect to x, then multiplying that answer by the derivative of x with respect to t, and finally adding that result to the product of the derivative of z with respect to y and the derivative of y with respect to t. In equation form, this can be written as:

\frac{dz}{dt} = \frac{dz}{dx}\frac{dx}{dt} + \frac{dz}{dy}\frac{dy}{dt}

In our case, we have that z = x2y2xyx sint , so we will take the derivative of each term in z separately. For the first term, x2y2 , we have:

\frac{d(x^2y^2)}{dx} = 2x(y^2) + (x^2)(2y)\frac{dy}{dx}

For the second term, xy , we have:

\frac{d(xy)}{dx} = y + x\frac{dy}{dx}

For the third term, x sint , we have:

\frac{d(x\sin t)}{dx} = \cos t + x\frac{dt}{dx}\sin t

### Find the integral of x^2 with respect to t

When finding the integral of a function with respect to t, we are looking for the anti-derivative of the function with respect to t. In this case, we are looking for the anti-derivative of x^2 with respect to t.

To find the integral of x^2 with respect to t, we can use the chain rule. The chain rule states that if we have a function h(x) that is a function of a function g(x), then we can find the derivative of h(x) with respect to x by multiply the derivative of g(x) with respect to x by the derivative of h(g(x)) with respect to g(x).

In this case, our function h(x) is x^2 and our function g(x) is t. We can find the derivative of g(t) with respect to t by taking the derivative of t (which is 1). We can find the derivative of h(g(t)) with respect to g(t) by taking the derivative of x^2 (which is 2x).

Therefore, when we use the chain rule, we get:

d/dx[x^2] = d/dt[t] * d/dx[x^2]

d/dx[x^2] = 1 * 2x

d/dx[x^2] = 2x

### Find the integral of y^2 with respect to t

To find the integral of y^2 with respect to t, we need to use the chain rule. The chain rule states that if we have a function z that is a function of x and y, and x and y are both functions of t, then we can find the derivative of z with respect to t by taking the derivative of z with respect to x and multiplying it by the derivative of x with respect to t, and then adding the derivative of z with respect to y multiplied by the derivative of y with respect to t. In other words, we have:

dzdt=dzdx⋅dxdt+dzdy⋅dydt

For our example, we have:

z=y2x2xyxsinty9et

So, we have:

dzdx=2yx2xyxsintyd9et

And: dzdy=x2(2y)xyxsintyd9et+x2y2xsint(siny)d9et+x2y26ytln(t)dt−1e−tcos(t)dt

Now we need to take the derivatives of x and y with respect to t. We have:

-xtan(t)−9e−tsin(t)

-ysiny+9ei−1e−tcot(t)+6ti−ltn(ti)+1

So, our final answer is: dzdt=tan(t)(2yx2xyxsintyd9et)+ysiny(x22yxylnsinx22yxylnsinsinfinal×termhere)+cot(ti)(x22yxylnsinx22yxylnsinynewfinaltermhere)+i⋅ltn⋅🡪tthenextfinaltermherei−52te5ln5termabovethisoneiln3iGoogleSiriBingSearchBarMarvinDictionaryThesaurusAppendixCitationsIndexSystemsOfEquationsAndInequalities1stQuadrantCoordinatePlaneGeometryTrigonometricRatiosSpecialAnglesCircularFunctionsSecantsTangentsAnglesIn radiansDegreesPlaneAnglesDirectMeasurementtrigonometricidentitiesUnitCircleInverseTrigonometricFunctionsFundamentalIdentitiesVerifyProvegraphsinhcoshtanhsecchccschcotcosechcschradianmeasureconventionsarc lengthlinear speedangular speeddistance around circlearclengthformulasdegreeseverythingsheetyeah!reviewcheck for understandingenrichmentunit circle applications4Q Coordinate PlaneGeometryGraph each equation on a coordinate plane. State the Quadrant in which the solutions lie.Rotate each figure about point (0, 0), (250px-right), through indicated angles. State coordinates for each vertex after rotation.State whether each triangle is similar or not similar to triangle ABC.(Rotate it if needed.)State whether line segment AB is parallel or perpendicular to line segment CD.(Flip/slide it if needed.)State whether angle AOB is acute, right or obtuse angle.(Rotate it if needed.)Name angles formed when lines intersect at point O Name angles formed when lines intersect at point OName sides opposite angles indicated 8th gradeThe coordinate plane can be divided into four quadrants by two perpendicular axes called the x-axis andthe y-axis as shown in Quadrant I below.An ordered pair (a, b) tells us where a point A lies onthe coordinate plane relativeto origin O. In other words, it tells us how far horizontally (a units alongthe x -axis from origin O)and how far vertically (b units along the y -axis from origin O)we must travel from originO until we reach A .• The first number in an ordered pair is always written firstand corresponds toreference point A’sposition alongthe horizontalaxis – thatis , its distancefrom originO horizontallyor along themillionsplace . Thisnumberiscalled anorderedpair’s” abscissa”• The second number corresponds toreference point A’sposition alongthey – axis ,or verticalposition – thatis , itsdistancefromoriginOvertically ordown onesagons place .This numberiscalled anorderedpair’s”ordinate “We will denotepoints on themap using theirordered pairs expressedas “( latitude ,longitude )”. Inthis notationlatitude always comesbefore longitude inthe ordered pair ,even thoughmost people wouldfirst say “45 degreesnorth latitude 150degrees eastlongitude .”Reference longitude lines are numbered eastward or westward fromthe prime meridian through 180° (+180° or − 180°). To calculateCoordinates we takethe signed anglebetween agiven referencelongitude lineto eachlongitudelinewestof thereferencelineand multiplyit per degreeby 360°/180° atthe equator;we then addthis resultto themeridianchildineastofthereferencelineand multiplyit per degreeby 360°/180° atthe equatorat 12■ hours ; addthis resultto thosehowed forthesamedistance southofthe equatorat 24 hours ;and subtractit for thosedistanceeastofthereferencedeterminedattime showedfor that listedtime zone southof GreenwichMean Time

### Find the integral of xy with respect to t

When finding the integral of a function, you are looking for the area underneath its curve. This area can be found by calculating the definite integral of the function. The definite integral is the limit of a summation, and it can be found using the chain rule.

The chain rule states that if you have a function f(x) that is composed of g(x) and h(x), then the derivative of f(x) with respect to x is: f'(x) = g'(x)*h(x) + h'(x)*g(x).

To find the definite integral of xy with respect to t, we can use the chain rule. First, we need to find the derivative of xy with respect to t. We can do this by finding the derivative of x with respect to t (which is 1), and multiplying it by y (derivative of y with respect to t is 0), and adding that to the derivative of y with respect to t (which is 1) multiplied by x. So, our final answer is: f'(t) = 1*y + x*1 = y + x.

Now that we have our derivative, we can use it to find our integral. An integral is just a summation, so we will take our derivative and sum it from t=0 to t=1. This gives us: integal[xy]dt = summation[y+x]dt from t=0 to t=1 = y + x evaluated at t=1 – y + x evaluated at t=0 = 1 + 0 – 0 – 1 = 0. So, our final answer is that the integral of xy wrt to t is 0.

### Find the integral of x with respect to t

The chain rule is a powerful tool that can be used to find integrals of functions that are composed of other functions. In the example above, we have a function z that is composed of two other functions, x and y. To find the integral of z with respect to t, we need to use the chain rule.

The chain rule tells us that we can find the derivative of a composite function by multiplying the derivative of the inner function by the derivative of the outer function. In our example, we have:

dzdt = dx dt + dy dt

To find the integral of x with respect to t, we need to integrate both sides of this equation. on the left hand side, we have:

∫dzdt = ∫dx dt + ∫dy dt

On the right hand side, we have:

z = x2 y2 xy x sint y 9et

We can now use the chain rule to find the integral of each term on the right hand side. For example, to find the integral of x2 with respect to t, we would use:

∫x2 dt = ∫(x2)(1) dt = x2 ∫1 dt = x2 t + C

### Find the integral of sin(t) with respect to t

The chain rule is a tool that allows us to find the derivative of a function that is composed of other functions. In other words, it allows us to take the derivative of a function that is not just a single variable. In order to use the chain rule, we need to have a function that is in the form:

f(g(x))

Where g(x) is some other function of x. We can then use the chain rule to find the derivative by using the following formula:

d/dx[f(g(x))] = f'(g(x)) * g'(x)

Let’s apply this to our example. We want to find the integral of sin(t) with respect to t. So, our function is in the form f(g(x)). In this case, f(x) = sin(x) and g(x) = t. We can then use the chain rule formula to find the derivative:

d/dt[sin(t)] = cos(t)*1 (Since the derivative of t is just 1.)

Therefore, we can conclude that the integral of sin(t) with respect to t is just cos(t).