Answer :
Sure! Let's find the derivative of the function [tex]\( j(x) = 50x^3 - 25x^2 \)[/tex] with respect to [tex]\( x \)[/tex].
### Step-by-step solution:
1. Identify the function and its terms:
- We have the function: [tex]\( j(x) = 50x^3 - 25x^2 \)[/tex].
- This is a polynomial function with [tex]\( x^3 \)[/tex] and [tex]\( x^2 \)[/tex] terms.
2. Differentiate each term individually:
- The derivative of a term [tex]\( ax^n \)[/tex] is given by the formula: [tex]\( a \times n \times x^{n-1} \)[/tex].
3. Differentiate the first term [tex]\( 50x^3 \)[/tex]:
- Applying the formula: [tex]\( a = 50 \)[/tex], [tex]\( n = 3 \)[/tex].
- The derivative is: [tex]\( 50 \times 3 \times x^{3-1} = 150x^2 \)[/tex].
4. Differentiate the second term [tex]\( -25x^2 \)[/tex]:
- Applying the formula: [tex]\( a = -25 \)[/tex], [tex]\( n = 2 \)[/tex].
- The derivative is: [tex]\( -25 \times 2 \times x^{2-1} = -50x \)[/tex].
5. Combine the derivatives:
- Add the derivatives of the individual terms together: [tex]\( 150x^2 - 50x \)[/tex].
Therefore, the derivative of [tex]\( j(x) = 50x^3 - 25x^2 \)[/tex] is [tex]\( 150x^2 - 50x \)[/tex].
This result gives us the rate of change of the function [tex]\( j(x) \)[/tex] with respect to [tex]\( x \)[/tex].
### Step-by-step solution:
1. Identify the function and its terms:
- We have the function: [tex]\( j(x) = 50x^3 - 25x^2 \)[/tex].
- This is a polynomial function with [tex]\( x^3 \)[/tex] and [tex]\( x^2 \)[/tex] terms.
2. Differentiate each term individually:
- The derivative of a term [tex]\( ax^n \)[/tex] is given by the formula: [tex]\( a \times n \times x^{n-1} \)[/tex].
3. Differentiate the first term [tex]\( 50x^3 \)[/tex]:
- Applying the formula: [tex]\( a = 50 \)[/tex], [tex]\( n = 3 \)[/tex].
- The derivative is: [tex]\( 50 \times 3 \times x^{3-1} = 150x^2 \)[/tex].
4. Differentiate the second term [tex]\( -25x^2 \)[/tex]:
- Applying the formula: [tex]\( a = -25 \)[/tex], [tex]\( n = 2 \)[/tex].
- The derivative is: [tex]\( -25 \times 2 \times x^{2-1} = -50x \)[/tex].
5. Combine the derivatives:
- Add the derivatives of the individual terms together: [tex]\( 150x^2 - 50x \)[/tex].
Therefore, the derivative of [tex]\( j(x) = 50x^3 - 25x^2 \)[/tex] is [tex]\( 150x^2 - 50x \)[/tex].
This result gives us the rate of change of the function [tex]\( j(x) \)[/tex] with respect to [tex]\( x \)[/tex].
