Math Problem Statement

Find the derivative of the function by the limit process in exercises 11-20.

Solution

You have provided a list of exercises asking to find the derivative of the function using the limit process. Let's go through some of these problems step-by-step using the limit definition of a derivative:

The limit definition of the derivative:

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

Let's solve a few of these:

11. f(x)=4f(x) = 4

This is a constant function. The derivative of any constant is 0. f(x)=0f'(x) = 0

13. f(x)=5xf(x) = -5x

This is a linear function. The derivative of a linear function axax is just the coefficient aa. f(x)=5f'(x) = -5

15. f(x)=x23f(x) = x^2 - 3

To apply the limit definition: f(x)=limh0(x+h)23(x23)hf'(x) = \lim_{h \to 0} \frac{(x+h)^2 - 3 - (x^2 - 3)}{h} Expanding and simplifying: f(x)=limh0x2+2xh+h23x2+3h=limh02xh+h2hf'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 3 - x^2 + 3}{h} = \lim_{h \to 0} \frac{2xh + h^2}{h} f(x)=limh0(2x+h)=2xf'(x) = \lim_{h \to 0} (2x + h) = 2x

17. f(x)=x312xf(x) = x^3 - 12x

Using the limit definition: f(x)=limh0(x+h)312(x+h)(x312x)hf'(x) = \lim_{h \to 0} \frac{(x+h)^3 - 12(x+h) - (x^3 - 12x)}{h} Expanding: f(x)=limh0x3+3x2h+3xh2+h312x12hx3+12xhf'(x) = \lim_{h \to 0} \frac{x^3 + 3x^2h + 3xh^2 + h^3 - 12x - 12h - x^3 + 12x}{h} Simplifying: f(x)=limh03x2h+3xh2+h312hh=limh0(3x2+3xh+h212)f'(x) = \lim_{h \to 0} \frac{3x^2h + 3xh^2 + h^3 - 12h}{h} = \lim_{h \to 0} (3x^2 + 3xh + h^2 - 12) f(x)=3x212f'(x) = 3x^2 - 12

Would you like me to solve more of these exercises? Let me know if you'd like further details or explanations.

Related Questions:

  1. How does the limit definition of the derivative work for a linear function like f(x)=ax+bf(x) = ax + b?
  2. What is the derivative of f(x)=xnf(x) = x^n for a general exponent nn?
  3. Why does the derivative of a constant function always equal zero?
  4. Can you apply the limit process for functions involving products, like f(x)=x2g(x)f(x) = x^2 \cdot g(x)?
  5. How do higher-order derivatives relate to the limit process?

Tip:

When computing derivatives using the limit definition, it's often helpful to expand all terms fully and cancel out similar ones before taking the limit.

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Math Problem Analysis

Mathematical Concepts

Derivatives
Limit process
Functions
Polynomials

Formulas

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

Theorems

Limit definition of a derivative

Suitable Grade Level

High school calculus