Math Problem Statement

Find the derivative of f(x)=x²+2x using the first principle.

Solution

The derivative of a function f(x)f(x) using the first principle (or definition of the derivative) is given by:

f(x)=limh0f(x+h)f(x)hf'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}

Given f(x)=x2+2xf(x) = x^2 + 2x, let's calculate the derivative step-by-step.

Step 1: Compute f(x+h)f(x+h)

Substitute x+hx+h into the function f(x)f(x):

f(x+h)=(x+h)2+2(x+h)f(x+h) = (x+h)^2 + 2(x+h) Expanding the terms: f(x+h)=(x2+2xh+h2)+(2x+2h)f(x+h) = (x^2 + 2xh + h^2) + (2x + 2h) f(x+h)=x2+2xh+h2+2x+2hf(x+h) = x^2 + 2xh + h^2 + 2x + 2h

Step 2: Set up the difference f(x+h)f(x)f(x+h) - f(x)

Now subtract f(x)=x2+2xf(x) = x^2 + 2x from f(x+h)f(x+h):

f(x+h)f(x)=(x2+2xh+h2+2x+2h)(x2+2x)f(x+h) - f(x) = (x^2 + 2xh + h^2 + 2x + 2h) - (x^2 + 2x) Simplify the expression: f(x+h)f(x)=x2+2xh+h2+2x+2hx22xf(x+h) - f(x) = x^2 + 2xh + h^2 + 2x + 2h - x^2 - 2x f(x+h)f(x)=2xh+h2+2hf(x+h) - f(x) = 2xh + h^2 + 2h

Step 3: Divide by hh

Next, divide the expression by hh:

f(x+h)f(x)h=2xh+h2+2hh\frac{f(x+h) - f(x)}{h} = \frac{2xh + h^2 + 2h}{h} Simplify: f(x+h)f(x)h=2x+h+2\frac{f(x+h) - f(x)}{h} = 2x + h + 2

Step 4: Take the limit as h0h \to 0

Finally, take the limit of the expression as h0h \to 0:

f(x)=limh0(2x+h+2)f'(x) = \lim_{h \to 0} (2x + h + 2) Since h0h \to 0, we get: f(x)=2x+2f'(x) = 2x + 2

Thus, the derivative of f(x)=x2+2xf(x) = x^2 + 2x using the first principle is:

f(x)=2x+2f'(x) = 2x + 2

Would you like more details or have any questions?

Relative questions:

  1. How is the first principle of differentiation different from using standard rules?
  2. Can you apply the first principle to find the derivative of f(x)=x3f(x) = x^3?
  3. What is the geometric interpretation of the derivative?
  4. How would the first principle work for non-polynomial functions, like sin(x)\sin(x)?
  5. What are the limitations of using the first principle for complex functions?

Tip:

The first principle of differentiation is a useful tool for understanding the concept of limits and how they apply to the slope of a function at a point.

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Math Problem Analysis

Mathematical Concepts

Calculus
Derivatives
First Principle of Differentiation

Formulas

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

Theorems

Definition of Derivative using First Principles

Suitable Grade Level

Grade 11-12