Answer: To find dy/dx of the given function y = x^3(4-3x+5x^2)^(1/2), we can apply the chain rule. Let's break down the process step by step:
First, let's define u as the function inside the parentheses: u = 4-3x+5x^2.
Next, we can rewrite the function as y = x^3u^(1/2).
Now, let's differentiate y with respect to x using the product rule and chain rule.
dy/dx = (d/dx)[x^3u^(1/2)]
Using the product rule, we have:
dy/dx = (d/dx)[x^3] * u^(1/2) + x^3 * (d/dx)[u^(1/2)]
Differentiating x^3 with respect to x gives us:
dy/dx = 3x^2 * u^(1/2) + x^3 * (d/dx)[u^(1/2)]
Now, we need to find (d/dx)[u^(1/2)] by applying the chain rule.
Let's define v as u^(1/2): v = u^(1/2).
Differentiating v with respect to x gives us:
(d/dx)[v] = (d/dv)[v^(1/2)] * (d/dx)[u]
= (1/2)v^(-1/2) * (d/dx)[u]
= (1/2)(4-3x+5x^2)^(-1/2) * (d/dx)[u]
Finally, substituting back into our expression for dy/dx:
dy/dx = 3x^2 * u^(1/2) + x^3 * (1/2)(4-3x+5x^2)^(-1/2) * (d/dx)[u]
Since (d/dx)[u] is the derivative of 4-3x+5x^2 with respect to x, we can calculate it separately:
(d/dx)[u] = (d/dx)[4-3x+5x^2]
= -3 + 10x
Substituting this back into the expression:
dy/dx = 3x^2 * u^(1/2) + x^3 * (1/2)(4-3x+5x^2)^(-1/2) * (-3 + 10x)
Simplifying further if desired, but this is the general expression for dy/dx based on the given function.
Explanation: