How do you find the derivative of 5x?

Answer 1

Let y = 5x

Then think y = 5x^(1)

Differentioation at its most simplistic is to move the exponential to be a coefficient. Then taking the original expoential subtract '1' from it , and use the answer as the new expoential .

Algebraically

y = Ax^(n)

dy/dx = Anx^(n-1)

So for the above example

y = 5x^(1)

dy/dx = 5(1)x^(1-1)

dy/dx = 5(1)x^(0)

Now any value to the power of '0' equa; '1'.

Hence

dy/dx = 5(1)(1)

dy/dx = 5 The answer.

Answer 2

In general, if you're taking the derivative with respect to X, then you take the current power of X, multiply the given quantity by that number and then subtract one from the current power. In this case, that's an overcomplicated way of describing what happens but here's the process: 5x is more fully 5*x^1 So you take the power (1) and multiply it it by the given quantity. This gives you 1*5*x^1 Now you subtract one from the current power giving you 1*5*x^0 which equals 5. So the answer is simply 5 in this case. But what if you were trying to find the derivative of 4x^7? In this case, you would multiply the quantity by 7 (giving you 7*4*x^7) and subtract 1 from the current power giving you a final answer of 28*x^6. This also works for negative powers and square roots. The derivative of sqrt(x) can be found by recognizing that this is equal to x^(1/2). So you multiply everything by 1/2 and subtract one from the power and get 1/2 * x^(-1/2) which equals 1/2 * 1/sqrt(x) = 1/(2*sqrt(x))