Find the derivative of the given function. Simplify and express the answer using positive exponents only.
y= ((9x^2-4)^3/7x^2+4)
y= uv^-1
y'= u'/v -uv'/v^2
where u=(9x^2-4)^3
u'=3 (9x^2-4)^2 (18x)
v=(7x^2+4)
v'=14x
Have fun with the algebra simplifying it.
To find the derivative of the given function, we can use the quotient rule and the chain rule of differentiation.
The quotient rule states that if we have a function in the form of (u/v), the derivative of that function is given by:
(d/dx) (u/v) = (v * du/dx - u * dv/dx) / v^2
Here, u = (9x^2 - 4)^3 and v = 7x^2 + 4.
Now, let's find du/dx and dv/dx.
Using the chain rule, du/dx is obtained by taking the derivative of the outer function with respect to the inner function, and then multiplying it by the derivative of the inner function. In this case, the outer function is raising to the power of 3 and the inner function is 9x^2 - 4.
So, du/dx = 3(9x^2 - 4)^(3-1) * d/dx(9x^2 - 4)
To find d/dx(9x^2 - 4), we can use the power rule of differentiation, which states that the derivative of x^n is nx^(n-1). Therefore, the derivative of 9x^2 - 4 is 18x.
Hence, du/dx = 3(9x^2 - 4)^2 * 18x.
Next, let's find dv/dx.
The derivative of v = 7x^2 + 4 can be found using the power rule of differentiation. Since there is no variable other than x in v, the derivative is simply the derivative of x^2, which is 2x.
Now that we have du/dx and dv/dx, we can find the derivative of y.
(d/dx) (y) = (v * du/dx - u * dv/dx) / v^2
= ((7x^2 + 4) * 3(9x^2 - 4)^2 * 18x - (9x^2 - 4)^3 * 2x) / (7x^2 + 4)^2
Simplifying this expression further is possible, but expressing the answer using positive exponents only may not be feasible. Nonetheless, we can simplify the expression by distributing and combining like terms in the numerator.
So, the derivative of y = ((7x^2 + 4) * 3(9x^2 - 4)^2 * 18x - (9x^2 - 4)^3 * 2x) / (7x^2 + 4)^2
Please note that further simplification may be possible depending on the specific values of x, but it may not necessarily yield an expression with positive exponents only.