Find derivative w/ respect to x:
y=x^5/(1-x)�ã(x^2+2)
(hint: there is an easy way and a hard way!)
To find the derivative of the given expression with respect to x, we can use the quotient rule. The quotient rule states that for a function u/v, the derivative is given by (v * du/dx - u * dv/dx) / v^2.
In this case, let's define u = x^5 and v = (1-x)^(x^2+2). We need to find du/dx and dv/dx.
To find du/dx, we can use the power rule for derivatives. The power rule states that the derivative of x^n with respect to x is n*x^(n-1). Applying the power rule, we have:
du/dx = 5x^(5-1) = 5x^4.
Now let's find dv/dx. To do this, we need to apply both the product rule and the chain rule. Define w = 1-x and z = x^2+2. We can rewrite v = w^z and differentiate it step by step.
dv/dx = dz/dx * dw/dx,
where dz/dx and dw/dx represent the derivatives of z and w, respectively.
Using the chain rule, we have:
dz/dx = d(x^2+2)/dx = 2x,
dw/dx = d(1-x)/dx = -1.
So, dv/dx = (2x) * (-1) = -2x.
Now we have du/dx = 5x^4 and dv/dx = -2x. Substituting these values into the quotient rule, we get:
dy/dx = (v * du/dx - u * dv/dx) / v^2
= [(1-x)^(x^2+2) * 5x^4 - x^5 * (-2x)] / [(1-x)^(x^2+2)]^2.
Simplifying further, we have:
dy/dx = [(1-x)^(x^2+2) * 5x^4 + 2x^6] / [(1-x)^(x^2+2)]^2.
To verify our answer, we can substitute specific values of x and compare the derivative value obtained using this formula with the derivative obtained using numerical methods or an online calculator.