# Find the derivative of the following function with respect to x. y = 5e^x−2xsin(x)

So, what I would be looking for from students who are answering this questions are the application of differentiation techniques that they would have been taught at Year 12. The first step in answering this question is that students know that they have to apply the product rule for differentiable functions. They would first need to use the rule for differentiating exponential functions to differentiate the first term of y which we will denote f = 5e^x then df/dx = 5e^x. As this function has only one x variable then we are not required to apply the product rule and can proceed. Then, differentiate -2xsin(X) with respect to x where students would be required to apply the product rule in order to differentiate this term. So let z = -2xsin(x) and dz/dx = vu'+uv' where u = -2x and v = sin(x). So we have split the original function into two separate functions of x as there are two functions of x which are multiplied together which would require us to apply the product rule. Then, we would have to calculate the values of u' and v' where u' and v' are the derivates of u and v respectively. So we would get that, u' = -2 and v' = cos(x). Furthermore, then substituting the values of u, u', v, v' into the product rule formula, you would get that dz/dx = 5e^x -2sin(x) -2xcos(x), where we have combined df/dx and dz/dx to get a final solution to dy/dx.

Answered by Niall H. Maths tutor

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