Question

Please help me! Find dy/dx if y=f(x)=((x-1)^3)(x-5)

Answer 1

To find the derivative of the function y = f(x) = ((x-1)^3)(x-5), we can use the product rule and the chain rule of differentiation. Let's break it down step by step:

Step 1: Expand the function We can simplify the function by expanding the expression (x-1)^3 using the binomial theorem: f(x) = (x^3 - 3x^2 + 3x - 1)(x - 5)

Step 2: Apply the product rule The product rule states that if we have a function u(x) multiplied by another function v(x), then the derivative of the product is given by: (d/dx)(u(x)v(x)) = u'(x)v(x) + u(x)v'(x)

In our case, we have u(x) = (x^3 - 3x^2 + 3x - 1) and v(x) = (x - 5). Let's find the derivatives of u(x) and v(x) separately:

u'(x) = d/dx (x^3 - 3x^2 + 3x - 1) = 3x^2 - 6x + 3

v'(x) = d/dx (x - 5) = 1

Step 3: Apply the product rule to find dy/dx Now, we can apply the product rule to find the derivative of f(x) = u(x)v(x): dy/dx = u'(x)v(x) + u(x)v'(x) = (3x^2 - 6x + 3)(x - 5) + (x^3 - 3x^2 + 3x - 1)(1)

Simplifying this expression will give us the final derivative.