Algebra of Derivatives | Calculus Tutorials

1.Find the derivative of x²+3x+5.

Answer

Solution

(x2+3x+5)’= (x2)’+3(x)’+0= 2x+3

2. Find the derivative of sin²(x) using the product rule.

Answer

1. 2 sin(x) cos(x)

Solution

1. sin2(x) = sin(x) sin(x). Using the product rule with f = g = sin(x), we get (sin(x) sin(x))’ =cos(x) sin(x) + sin(x) cos(x) = 2 sin(x) cos(x).

3.Find the derivative of sin²(x) using the chain rule, with g = sin(x).

Answer

1. 2 sin(x) cos(x)

Solution

1. If we set g = sin(x), then f = g². Using the chain rule gives us: f(g)’ = 2 g g’ = 2 sin(x) cos(x)

4.Find the derivative of √(1+2x).

Answer

1. (1 + 2x)^-1/2

Solution

1. We will use the product rule with f = x, g = e^2x: (x ⋅ e^2x)’ = 1 ⋅ e^2x + x ⋅ 2e^2x = (1 + x) e^2x

5.Find the derivative of x e^2x

Answer

1. (1 + x) e^2x

Solution

1. We will use the product rule with f = x, g = e^2x: (x ⋅ e^2x)’ = 1 ⋅ e^2x + x ⋅ 2e^2x = (1 + x)e^2x