How do you differentiate y = 5 x^3 + 1/2 x^2 + 3x -4

FIrslty to avoid confusion:

x = the variable x

X = multiplication.

When you differentiate an equation, the derivative (answer) is referred to as dy/dx.

For a polynomial as in this question you take each term individually and differentiate that.

Let's first do this for a generic term axb and we will then apply it to thisquestion​​:

The formula for the derivative of axb is baxb-1​, which means you take the degree of x (whatever power it is being raised to, which in this case is b) and multiply it by the coefficient of x (which in this case in a) which provides the ba part of the solution. You then reduce the degree of x by one so it becomes xb-1 instead of just xb. Therefore when you put the two parts together you get the final derivative as baxb-1.​

Now let's look at the first term of our equation : 5x3​.

The degree here is 3 so the first thing you do is multiply 3 by the coefficient of x which is 5 giving you 3x5 which is 15​.

Next, you reduce the degree of x by 1. It was x3 so now it is x3-1 which is x2​.

Now you can put the two parts together to give the derivative of 5x3 being 15x2.

If you do the same thing for the next term, then you get (2x1/2)x2-1​ which is x.

For the third time the derivative is quite nice, if you just have a number followed by x such as 3x as we have here, to get the derivative you just remove the x, meaning the derivative is 3.

For the final term of the equation we just have a number. Whenever you differentiate a number on its own, the derivative is 0.

So now the final thing left to do is put our answer together. You need to take each of our 4 derivatives and simply add them together.

This would give you: 15x2​ + x + 3 + 0

Meaning the final answer dy/dx = 15x2​ + x + 3.

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