Question
Problem and solution on derivatives.
And easy steps and understanding
And easy steps and understanding
Answers
GPT-4o mini
Sure! Let's break down the concept of derivatives in calculus, provide a basic problem, and then work through a solution step-by-step.
### What are Derivatives?
In calculus, the derivative of a function measures how the function's output value changes as its input value changes. In simpler terms, it tells us the rate of change of a function at any given point.
**Notation**:
- The derivative of a function \( f(x) \) is often denoted as \( f'(x) \) or \( \frac{df}{dx} \).
### Basic Rule
One of the most commonly used rules for differentiation is the power rule:
- If \( f(x) = x^n \) (where \( n \) is any real number), then
\[
f'(x) = nx^{n-1}.
\]
### Problem
Let's find the derivative of the function:
\[
f(x) = 3x^4 + 5x^2 - 2.
\]
### Steps to Solve the Derivative
1. **Identify the function**:
Here, we have \( f(x) = 3x^4 + 5x^2 - 2 \).
2. **Separate the terms**:
Break down the function into individual terms that can be differentiated using basic rules:
- \( 3x^4 \)
- \( 5x^2 \)
- \( -2 \) (which is a constant)
3. **Differentiate each term**:
- For \( 3x^4 \):
Using the power rule:
\[
f'(x) = 3 \cdot 4x^{4 - 1} = 12x^3.
\]
- For \( 5x^2 \):
Again using the power rule:
\[
f'(x) = 5 \cdot 2x^{2 - 1} = 10x.
\]
- For the constant \( -2 \):
The derivative of any constant is \( 0 \).
4. **Combine the derivatives**:
Adding the derivatives of each term together, we get:
\[
f'(x) = 12x^3 + 10x + 0.
\]
Simplifying gives us:
\[
f'(x) = 12x^3 + 10x.
\]
### Solution
Thus, the derivative of the function \( f(x) = 3x^4 + 5x^2 - 2 \) is:
\[
f'(x) = 12x^3 + 10x.
\]
### Summary of Steps:
1. Identify the function you want to differentiate.
2. Break it down into individual terms if it is a polynomial.
3. Apply the power rule to each term.
4. Combine the results to get the final derivative.
This approach provides a structured way to tackle differentiation problems. The key is to remember the rules and practice. If you have more specific types of problems or concepts you want to cover, feel free to ask!
### What are Derivatives?
In calculus, the derivative of a function measures how the function's output value changes as its input value changes. In simpler terms, it tells us the rate of change of a function at any given point.
**Notation**:
- The derivative of a function \( f(x) \) is often denoted as \( f'(x) \) or \( \frac{df}{dx} \).
### Basic Rule
One of the most commonly used rules for differentiation is the power rule:
- If \( f(x) = x^n \) (where \( n \) is any real number), then
\[
f'(x) = nx^{n-1}.
\]
### Problem
Let's find the derivative of the function:
\[
f(x) = 3x^4 + 5x^2 - 2.
\]
### Steps to Solve the Derivative
1. **Identify the function**:
Here, we have \( f(x) = 3x^4 + 5x^2 - 2 \).
2. **Separate the terms**:
Break down the function into individual terms that can be differentiated using basic rules:
- \( 3x^4 \)
- \( 5x^2 \)
- \( -2 \) (which is a constant)
3. **Differentiate each term**:
- For \( 3x^4 \):
Using the power rule:
\[
f'(x) = 3 \cdot 4x^{4 - 1} = 12x^3.
\]
- For \( 5x^2 \):
Again using the power rule:
\[
f'(x) = 5 \cdot 2x^{2 - 1} = 10x.
\]
- For the constant \( -2 \):
The derivative of any constant is \( 0 \).
4. **Combine the derivatives**:
Adding the derivatives of each term together, we get:
\[
f'(x) = 12x^3 + 10x + 0.
\]
Simplifying gives us:
\[
f'(x) = 12x^3 + 10x.
\]
### Solution
Thus, the derivative of the function \( f(x) = 3x^4 + 5x^2 - 2 \) is:
\[
f'(x) = 12x^3 + 10x.
\]
### Summary of Steps:
1. Identify the function you want to differentiate.
2. Break it down into individual terms if it is a polynomial.
3. Apply the power rule to each term.
4. Combine the results to get the final derivative.
This approach provides a structured way to tackle differentiation problems. The key is to remember the rules and practice. If you have more specific types of problems or concepts you want to cover, feel free to ask!