Can someone show me how this equation works and who you get through each step and what does dy/dx mean

y=3(2x+1)^3-1

dy/dx = 3(3)(2x+1)^2 (2)
dy/dx = 18(2x+1)^2

d^2y/dx^2 = (18)(2)(2x+1)(2)
d^2y/dx^2 = 36(2x+1)

Hello;
dy/dx means to find the derivative of the function ' y ' with respect to ' x '.

In order to understand how the derivation of y=3(2x+1)^3-1 is done, need to know these derivation rules:

1) d[u^n]/dx = n*[u^(n-1)]*(du/dx) where u: Any function of x
n: The exponent of u.

Here, in your case, u^n = (2x + 1)^3 thus, d[u^n]/dx = n*[u^(n-1)]*(du/dx) = 3*(2x + 1)^2*(2)

2) d(constant)/dx = ZERO
3) d(a - b)/dx = d(a)/dx - d(b)/dx [The derivative of the difference is equals to the difference of the derivative]

Now let us apply these formulas on your case:

y=3(2x+1)^3-1


dy/dx = d[3(2x+1)^3-1]/dx = d[3(2x+1)^3]/dx - d[1]/dx [ By applying the third formula]

= 3*3*(2x + 1)^2*(2) - 0 [ By applying the first and the second formulas ]

= 18*(2x + 1)^2

The same formula can be applied for finding d^2y/dx^2


hope that is clear, Bye

these are calculus working dy/dx is shorthand reminding us of what the derivative is it is the slope of an equation dy/dx stands for delta y divided by delta x the way we find derivatives what someone has done here is apply the chain rule to find the first and second derivatives they made a slight mistake on the last line but that is not uncommon another way to write dy/dx is f' see the little mark after the f it means f prime which is the same as first derivative of f

Further,
d^3y/dx^3 = 36 (1)(2x+1)^0(2)
d^3y/dx^3 = 36(1)(1)(2)
d^3y/dx^3 = 72 or y'''=72 (Answer)

dy/dx mean that to take derivate of y with respect to x. In this equation Y be a dependent variable and X be an independent variable.

y=3(2x+1)^3-1 put u=2x+1 so du/dx=2

y=3u^3-1

dy/du=9u^2

dy/dx=dy/du*du/dx= 9u^2*2=18u^2=18(2x+1)^2 dy/dx is 1 st derivative d^2y/d^2x is 2 nd derivative