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Function evaluation is the process of finding the value of a function for a given input. It involves substituting the given value into the function's equation. In our example, the function is given by:

f(x) = x^{1/3}

To evaluate the function at x = 8, substitute 8 into the function:

f(8) = 8^{1/3}

Here, we need to find the cube root of 8, which is 2. Therefore, f(8) = 2. This process of substitution is essential in understanding how functions behave for different inputs.

• Substitute means replacing the variable with a specific value.
• For simple functions like polynomials, it is straightforward.
• Always follow the order of operations: Parentheses, Exponents, Multiplication and Division, Addition and Subtraction (PEMDAS).

The power rule is a basic rule in calculus for finding derivatives. It states that if you have a function of the form:

f(x) = x^n

The derivative of this function is given by:

f'(x) = n * x^{n-1}

Let's apply this to our function f(x) = x^{1/3}:

f(x) = x^{1/3}

Using the power rule, we get:

f'(x) = (1/3) * x^{(1/3)-1}

Simplify the exponent:

f'(x) = (1/3) * x^{-2/3}

Understanding this rule helps to differentiate polynomial functions quickly and efficiently.

• The power n is moved to the front as a coefficient.
• The exponent is reduced by 1.
• Practice with various n values to get comfortable.

Now that we have the derivative of our function f(x), we can find the value of the derivative at a specific point. This is useful to understand the rate of change of the function at that point.

Let's find f'(8) for our function:

f'(x) = (1/3) * x^{-2/3}

Substitute x = 8 into the derivative:

f'(8) = (1/3) * 8^{-2/3}

To simplify, first find 8^{2/3}. Notice that 8 is the same as 2^3:

8^{2/3} = (2^3)^{2/3} = 2^2 = 4

Plug this back into the derivative:

f'(8) = (1/3) * (1/4) = 1/12

Thus, f'(8) = 1/12.

• Substituting the value of x into the derivative.
• Simplifying the expression by breaking it down into steps.
• Understanding how changes in x affect the function's rate of change.