Now, use the Power Rule to take the derivative of each term. The derivative of x = x . Thederivative of x = x . Therefore, the derivative is = .SOLUTIONS TO PRACTICE PROBLEM SET 5
1.
We find the derivative using the Quotient Rule, which says that if f(x) = , then f′(x) = . Here, f(x) = , so u = 4x^3 − 3x^2 and v = 5x^7 +. Using the Quotient Rule,
we get f′(x) = . This can be simplified to f′(x) = .- 3 x^2 − 6x − 1
We find the derivative using the Product Rule, which says that if f(x) = uv, then f′(x) = u + v. Here f(x) = (x^2 − 4x + 3)(x + 1), so u = x^2 − 4x + 3 and v = x + 1. Using the Product Rule,
we get f′(x) = (x^2 − 4x + 3)(1) + (x + 1)(2x − 4). This can be simplified to f′(x) = 3x^2 − 6x − 1.3.
We find the derivative using the Chain Rule, which says that if y = f(g(x)), then y′ = . Here f(x) = 8 , which can be written as f(x) = 8(x^4 − 4x^2 ). Using
the Chain Rule, we get f′(x) = 8 (x^4 − 4x^2 ) (4x^3 − 8x). This can be simplified to f′(x) =