derivative of y=sqrt2x-3xy+2
Is the function
y=sqrt(2x)-3xy+2
or
y=sqrt(2x-3xy+2)
Parentheses are very important in math.
in either case, apply the chain rule.
If u is a function of x, then if y is a function of u, we have
dy/dx = dy/du * du/dx
so, what do you get?
To find the derivative of the given function y = √(2x) - 3xy + 2, we will use the rules of differentiation step by step.
Step 1: Differentiate each term separately.
The first term is √(2x). To find its derivative, we will use the chain rule, where we differentiate the outer function first and then multiply it by the derivative of the inner function.
Let's call the inner function u = 2x. Then, the outer function is f(u) = √u. Now we can differentiate it:
df/du = 1/(2√u)
Now let's differentiate the inner function:
du/dx = 2
Using the chain rule, we can multiply these two derivatives:
dy/dx = (df/du) * (du/dx) = (1/(2√u)) * 2 = 1/√(2x)
So the derivative of √(2x) is 1/√(2x).
Step 2: Differentiate the second term in the same way:
dy/dx = -3xy
To differentiate this, we will apply the product rule. The product rule states that if we have a function h(x) that is the product of two functions u(x) and v(x), then the derivative of h(x) is given by:
dh/dx = u(x) * dv/dx + v(x) * du/dx
In this case, u(x) = -3x and v(x) = y. We know that du/dx = -3 and dv/dx = dy/dx. Plugging these values into the product rule, we get:
dy/dx = u(x) * dv/dx + v(x) * du/dx
= -3x * dy/dx + y * (-3)
Step 3: Differentiate the constant term:
The derivative of a constant term is always 0. Since the last term is 2, its derivative is 0.
Now we can rewrite our result from step 2:
dy/dx = -3x * dy/dx - 3y
Step 4: Solve for dy/dx:
To solve for dy/dx, we can rearrange the equation:
dy/dx + 3x * dy/dx = -3y
Next, we can factor out the dy/dx term:
(1 + 3x) * dy/dx = -3y
Finally, divide both sides of the equation by (1 + 3x):
dy/dx = -3y / (1 + 3x)
So the derivative of the function y = √(2x) - 3xy + 2 is dy/dx = -3y / (1 + 3x).