Find the derivative of the function. y = sin(x) ln(8 + 3v) dv cos(x)

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Naomi Little eNotes educator | Certified Educator

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First, it depends on how you interpret the `dv` term. The most obvious way is to interpret it as `(dv)/dx` which is how I'll treat it. There are two things to remember with regard to differential calculus when looking at this question: the product rule and the chain rule:

`d/(dx) (UV) = (dU)/dx * V + (dV)/dx * U` - Product rule

`d/dx F(g(x)) = (dF)/(dg) * (dg)/(dx)` - Chain rule

The product rule comes into play when we see the above as a set of 4 functions of x multiplied together:

`f_1(x) = sin(x)`

`f_2(x) = ln(8+3v)`

`f_3(x) = (dv)/(dx)`

`f_4(x) = cos(x)`

Which means the following about their derivatives:

`d/dx f_1 = cos(x)`

`d/dxf_2 = 1/(8+3v) * 3 * (dv)/(dx)`

`d/dxf_3=(d^2v)/(dx^2)`

`d/dx(f_4) = -sin(x)`

Note, for `f_2` you must first interpret it...

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