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Problem and solution on derivatives
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GPT-4o mini
Sure! Let's start with a brief overview of derivatives and then I’ll provide you with a few easy problems along with their solutions.
### What is a Derivative?
The derivative of a function measures how the function's output value changes as its input changes. In simpler terms, it represents the rate of change or the slope of the function at any given point.
### Basic Rule
The derivative of a function \( f(x) \) can be denoted as \( f'(x) \) or \( \frac{df}{dx} \). A common rule is:
- For a power function \( f(x) = x^n \), the derivative is \( f'(x) = nx^{n-1} \).
### Example Problem 1
**Problem:** Find the derivative of the function \( f(x) = 3x^4 + 5x^2 - 2x + 7 \).
**Solution:**
To find the derivative, we apply the power rule:
- The derivative of \( 3x^4 \) is \( 12x^3 \).
- The derivative of \( 5x^2 \) is \( 10x \).
- The derivative of \( -2x \) is \( -2 \).
- The derivative of a constant \( 7 \) is \( 0 \).
So, combining these, we get:
\[ f'(x) = 12x^3 + 10x - 2. \]
---
### Example Problem 2
**Problem:** Find the derivative of the function \( g(x) = \frac{1}{2}x^2 - 3x + 4 \).
**Solution:**
Using the power rule:
- The derivative of \( \frac{1}{2}x^2 \) is \( x \).
- The derivative of \( -3x \) is \( -3 \).
- The derivative of \( 4 \) is \( 0 \).
Combining these, we have:
\[ g'(x) = x - 3. \]
---
### Example Problem 3
**Problem:** If \( h(t) = t^3 + 2t^2 - 5t + 1 \), find \( h'(t) \).
**Solution:**
Again using the power rule:
- The derivative of \( t^3 \) is \( 3t^2 \).
- The derivative of \( 2t^2 \) is \( 4t \).
- The derivative of \( -5t \) is \( -5 \).
- The derivative of constant \( 1 \) is \( 0 \).
Thus, we get:
\[ h'(t) = 3t^2 + 4t - 5. \]
---
### Example Problem 4
**Problem:** Calculate the derivative of \( k(x) = 4 - x^3 + 6x \).
**Solution:**
Using the power rule:
- The derivative of \( 4 \) is \( 0 \).
- The derivative of \( -x^3 \) is \( -3x^2 \).
- The derivative of \( 6x \) is \( 6 \).
Combining these results gives us:
\[ k'(x) = -3x^2 + 6. \]
---
### Example Problem 5
**Problem:** Find the derivative of \( f(x) = 5x^5 - 10x^3 + 2 \).
**Solution:**
Using the power rule:
- The derivative of \( 5x^5 \) is \( 25x^4 \).
- The derivative of \( -10x^3 \) is \( -30x^2 \).
- The derivative of \( 2 \) is \( 0 \).
So, we have:
\[ f'(x) = 25x^4 - 30x^2. \]
---
Feel free to try these problems on your own, and you can always come back if you'd like more practice or have questions!
### What is a Derivative?
The derivative of a function measures how the function's output value changes as its input changes. In simpler terms, it represents the rate of change or the slope of the function at any given point.
### Basic Rule
The derivative of a function \( f(x) \) can be denoted as \( f'(x) \) or \( \frac{df}{dx} \). A common rule is:
- For a power function \( f(x) = x^n \), the derivative is \( f'(x) = nx^{n-1} \).
### Example Problem 1
**Problem:** Find the derivative of the function \( f(x) = 3x^4 + 5x^2 - 2x + 7 \).
**Solution:**
To find the derivative, we apply the power rule:
- The derivative of \( 3x^4 \) is \( 12x^3 \).
- The derivative of \( 5x^2 \) is \( 10x \).
- The derivative of \( -2x \) is \( -2 \).
- The derivative of a constant \( 7 \) is \( 0 \).
So, combining these, we get:
\[ f'(x) = 12x^3 + 10x - 2. \]
---
### Example Problem 2
**Problem:** Find the derivative of the function \( g(x) = \frac{1}{2}x^2 - 3x + 4 \).
**Solution:**
Using the power rule:
- The derivative of \( \frac{1}{2}x^2 \) is \( x \).
- The derivative of \( -3x \) is \( -3 \).
- The derivative of \( 4 \) is \( 0 \).
Combining these, we have:
\[ g'(x) = x - 3. \]
---
### Example Problem 3
**Problem:** If \( h(t) = t^3 + 2t^2 - 5t + 1 \), find \( h'(t) \).
**Solution:**
Again using the power rule:
- The derivative of \( t^3 \) is \( 3t^2 \).
- The derivative of \( 2t^2 \) is \( 4t \).
- The derivative of \( -5t \) is \( -5 \).
- The derivative of constant \( 1 \) is \( 0 \).
Thus, we get:
\[ h'(t) = 3t^2 + 4t - 5. \]
---
### Example Problem 4
**Problem:** Calculate the derivative of \( k(x) = 4 - x^3 + 6x \).
**Solution:**
Using the power rule:
- The derivative of \( 4 \) is \( 0 \).
- The derivative of \( -x^3 \) is \( -3x^2 \).
- The derivative of \( 6x \) is \( 6 \).
Combining these results gives us:
\[ k'(x) = -3x^2 + 6. \]
---
### Example Problem 5
**Problem:** Find the derivative of \( f(x) = 5x^5 - 10x^3 + 2 \).
**Solution:**
Using the power rule:
- The derivative of \( 5x^5 \) is \( 25x^4 \).
- The derivative of \( -10x^3 \) is \( -30x^2 \).
- The derivative of \( 2 \) is \( 0 \).
So, we have:
\[ f'(x) = 25x^4 - 30x^2. \]
---
Feel free to try these problems on your own, and you can always come back if you'd like more practice or have questions!