A Comprehensive Guide to Finding the Formal Derivative of a Function at a Specific Point

Formal derivative form at x=a

To find the formal derivative of a function at a specific point, x=a, we need to apply certain rules and formulas

To find the formal derivative of a function at a specific point, x=a, we need to apply certain rules and formulas.

The formal derivative of a function f(x) is denoted as f'(x), and it represents the rate at which the function is changing at any given value of x.

To find the formal derivative of a function at x=a, we will use the definition of the derivative, which is based on the concept of limits. The derivative is defined as the limit of the difference quotient as the change in x becomes infinitesimally small:

f'(a) = lim (h -> 0) [f(a+h) – f(a)] / h

Here, h represents a small change in x.

To find the formal derivative, we will follow these steps:

Step 1: Substitute x=a in the given function f(x) and simplify the expression. We will replace all instances of x in the function with a.

Step 2: Apply the limit as h approaches 0 to the simplified expression obtained in the first step. This will involve evaluating the numerator and denominator separately.

Step 3: Simplify the resulting expression obtained in step 2 and obtain the formal derivative.

It’s important to note that the process is different for different types of functions. Depending on the function, we may need to use various derivative rules such as power rule, product rule, quotient rule, chain rule, etc.

Here is an example to illustrate the process:

Example: Find the formal derivative of the function f(x) = 2x^3 – 5x^2 + 3x at x=2.

Step 1: Substitute x=2 in the given function:
f(2) = 2(2)^3 – 5(2)^2 + 3(2) = 2(8) – 5(4) + 6 = 16 – 20 + 6 = 2

Step 2: Apply the limit as h approaches 0 to the difference quotient:
f'(2) = lim (h -> 0) [f(2+h) – f(2)] / h

Substituting f(2) = 2, we get:
f'(2) = lim (h -> 0) [f(2+h) – 2] / h

Step 3: Simplify the expression obtained in step 2:
f'(2) = lim (h -> 0) [(2(2+h)^3 – 5(2+h)^2 + 3(2+h)) – 2] / h

Expanding and simplifying the expression:
f'(2) = lim (h -> 0) [(2(8 + 12h + 6h^2 + h^3) – 5(4 + 4h + h^2) + 6 + 3h) – 2] / h
= lim (h -> 0) [(16 + 24h + 12h^2 + 2h^3 – 20 – 20h – 5h^2 + 6 + 3h – 2) / h]
= lim (h -> 0) [(2h^3 + 7h^2 + 7h)] / h
= lim (h -> 0) [h(2h^2 + 7h + 7)] / h
= lim (h -> 0) (2h^2 + 7h + 7)

Evaluating the expression as h approaches 0:
f'(2) = 2(0)^2 + 7(0) + 7 = 7

Therefore, the formal derivative of the function f(x) = 2x^3 – 5x^2 + 3x at x=2 is 7.

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