Hi, please help! How do i do this?
differentiate (2ax^2 + bx)/(bx^3 - cx)
Bailey, Andrea, Danny, Riley, anabelle, holly, LILLY, zachary, I'm stumped -- or whoever!
To quote one of our very good math and science tutors: “You will find here at Jiskha that long series of questions, posted with no evidence of effort or thought by the person posting, will not be answered. We will gladly respond to your future questions in which your thoughts are included.”
Treat the function as the ratio
f(x) = u(x)/v(x)
and use the formula for the derivative of the ratio of two functions
df/dx = [v du/dx - u dv/dx ]/ v^2
The rest is easy.
To differentiate the expression (2ax^2 + bx)/(bx^3 - cx), we can start by applying the quotient rule, as the expression is a ratio of two functions. The quotient rule states that the derivative of the quotient of two functions f(x) and g(x) is given by the formula:
(f'(x) * g(x) - f(x) * g'(x)) / (g(x))^2
Let's break down the problem to solve it step by step:
Step 1: Find the derivative of the numerator, f(x).
The numerator is 2ax^2 + bx.
To differentiate 2ax^2 with respect to x, we can use the power rule, which states that the derivative of x^n with respect to x is n * x^(n-1), where n is a constant.
So, differentiating 2ax^2 gives us 4ax.
Next, differentiating bx results in just b, as the derivative of a constant with respect to x is zero.
So, f'(x) = 4ax + b.
Step 2: Find the derivative of the denominator, g(x).
The denominator is bx^3 - cx.
Differentiating bx^3 with respect to x gives us 3bx^2, again using the power rule.
Differentiating -cx with respect to x gives us -c, as the derivative of a constant with respect to x is zero.
So, g'(x) = 3bx^2 - c.
Step 3: Apply the quotient rule.
Now that we have f'(x) and g'(x), we can apply the quotient rule formula:
(f'(x) * g(x) - f(x) * g'(x)) / (g(x))^2
Substituting the values:
= [(4ax + b) * (bx^3 - cx) - (2ax^2 + bx) * (3bx^2 - c)] / (bx^3 - cx)^2
Step 4: Simplify the expression.
To simplify the expression further, you can expand the products and rearrange the terms. After simplification, the final result should be the derivative of the given expression.
Keep in mind that this explanation covers the process of finding the derivative of the given expression. If you would like the final simplified expression, please provide the specific values of a, b, and c.