Answer :
Let's find the derivatives of the functions [tex]\( f(x) = 4x^7 \)[/tex] and [tex]\( g(x) = 7x^4 \)[/tex].
Step-by-step Solution:
1. Identify the Functions:
- [tex]\( f(x) = 4x^7 \)[/tex]
- [tex]\( g(x) = 7x^4 \)[/tex]
2. Apply the Power Rule for Derivatives:
The power rule states that if you have a function [tex]\( x^n \)[/tex], its derivative is [tex]\( nx^{n-1} \)[/tex].
3. Find the Derivative of [tex]\( f(x) = 4x^7 \)[/tex]:
- According to the power rule, [tex]\( x^7 \)[/tex] becomes [tex]\( 7x^6 \)[/tex] when differentiated.
- Multiply by the constant coefficient 4:
[tex]\[ \frac{d}{dx}(4x^7) = 4 \times 7x^6 = 28x^6 \][/tex]
4. Find the Derivative of [tex]\( g(x) = 7x^4 \)[/tex]:
- Using the power rule, [tex]\( x^4 \)[/tex] becomes [tex]\( 4x^3 \)[/tex] when differentiated.
- Multiply by the constant coefficient 7:
[tex]\[ \frac{d}{dx}(7x^4) = 7 \times 4x^3 = 28x^3 \][/tex]
Therefore, the derivatives are:
- [tex]\( \frac{d}{dx}(4x^7) = 28x^6 \)[/tex]
- [tex]\( \frac{d}{dx}(7x^4) = 28x^3 \)[/tex]
The correct answers are:
[tex]\[ 28x^6 \quad \text{and} \quad 28x^3 \][/tex]
Step-by-step Solution:
1. Identify the Functions:
- [tex]\( f(x) = 4x^7 \)[/tex]
- [tex]\( g(x) = 7x^4 \)[/tex]
2. Apply the Power Rule for Derivatives:
The power rule states that if you have a function [tex]\( x^n \)[/tex], its derivative is [tex]\( nx^{n-1} \)[/tex].
3. Find the Derivative of [tex]\( f(x) = 4x^7 \)[/tex]:
- According to the power rule, [tex]\( x^7 \)[/tex] becomes [tex]\( 7x^6 \)[/tex] when differentiated.
- Multiply by the constant coefficient 4:
[tex]\[ \frac{d}{dx}(4x^7) = 4 \times 7x^6 = 28x^6 \][/tex]
4. Find the Derivative of [tex]\( g(x) = 7x^4 \)[/tex]:
- Using the power rule, [tex]\( x^4 \)[/tex] becomes [tex]\( 4x^3 \)[/tex] when differentiated.
- Multiply by the constant coefficient 7:
[tex]\[ \frac{d}{dx}(7x^4) = 7 \times 4x^3 = 28x^3 \][/tex]
Therefore, the derivatives are:
- [tex]\( \frac{d}{dx}(4x^7) = 28x^6 \)[/tex]
- [tex]\( \frac{d}{dx}(7x^4) = 28x^3 \)[/tex]
The correct answers are:
[tex]\[ 28x^6 \quad \text{and} \quad 28x^3 \][/tex]
