Find x and y from the following? 2x+3=y+6 -4x-12=-8y
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Don't be scared with the t attached at the end.
This is known as the dummy variable. It's simply an arbitrary variable that represents some value at a particular time.
The next section is the derivative-integral relationship. They are simply inverses of each other; that is, when multiplied together, the two will cancel each other out; much like squares and square roots, as well as exponentials and logarithms.
So, we can simply see the derivative will be sinx + C.
Now, since the limit of the integral are two functions, we can apply the chain rule to them to get our derivative of the integral.
![\frac{d(\int_{f(x)}^{g(x)} sint\,dt)}{dx} = f'(x) \cdot sin[f(x)] - g'(x) \cdot sin[g(x)]](https://tex.z-dn.net/?f=%5Cfrac%7Bd%28%5Cint_%7Bf%28x%29%7D%5E%7Bg%28x%29%7D%20sint%5C%2Cdt%29%7D%7Bdx%7D%20%3D%20f%27%28x%29%20%5Ccdot%20sin%5Bf%28x%29%5D%20-%20g%27%28x%29%20%5Ccdot%20sin%5Bg%28x%29%5D)
![\frac{d(\int_{x^{3}}^{2x} sint\,dt)}{dx} = 3x^{2} \cdot sin[x^{3}] - 2 \cdot sin[2x]](https://tex.z-dn.net/?f=%5Cfrac%7Bd%28%5Cint_%7Bx%5E%7B3%7D%7D%5E%7B2x%7D%20sint%5C%2Cdt%29%7D%7Bdx%7D%20%3D%203x%5E%7B2%7D%20%5Ccdot%20sin%5Bx%5E%7B3%7D%5D%20-%202%20%5Ccdot%20sin%5B2x%5D)
This is known as the dummy variable. It's simply an arbitrary variable that represents some value at a particular time.
The next section is the derivative-integral relationship. They are simply inverses of each other; that is, when multiplied together, the two will cancel each other out; much like squares and square roots, as well as exponentials and logarithms.
So, we can simply see the derivative will be sinx + C.
Now, since the limit of the integral are two functions, we can apply the chain rule to them to get our derivative of the integral.