Differentiation is a mathematical process used to find the rate at which one quantity changes with respect to another. It is commonly used in calculus to find the slope of a curve at a specific point.
To differentiate a polynomial function, you can use the power rule, which states that the derivative of xn is nxn-1. For example, the derivative of 2x3 is 6x2.
The chain rule is a rule used to differentiate composite functions, which are functions that are made up of two or more functions. It states that the derivative of f(g(x)) is equal to the derivative of f evaluated at g(x) multiplied by the derivative of g(x).
To differentiate a function with a fractional exponent, you can use the power rule for fractional exponents, which states that the derivative of xa/b is equal to (a/b)xa/b-1. For example, the derivative of x3/2 is (3/2)x1/2.
Step 1: Use the power rule to differentiate 2x1 = 2.
Step 2: Use the chain rule to differentiate (x+4)^8 = 8(x+4)7.
Step 3: Simplify the expression to 2(8)(x+4)7 = 16(x+4)7.
Step 4: Use the quotient rule to differentiate x/(x+2) = (x+2)(1) - x(1) / (x+2)2 = (x+2-x) / (x+2)2 = 2 / (x+2).
Step 5: Use the chain rule to differentiate (x+2)^2 = 2(x+2)(1) = 2(x+2).
Step 6: Simplify the expression to 2 / 2(x+2) = 1 / (x+2).