Please provide the function for which you would like to find the derivative using the Chain Rule and other differentiation rules.
To find the derivative of the function \( y = e^{7x} (7x + 3)^4 \), we will use the product rule and the chain rule.
1. **Identify parts of the function**:
- Let \( u = e^{7x} \)
- Let \( v = (7x + 3)^4 \)
2. **Differentiate \( u \)**:
\[
u' = \frac{d}{dx}(e^{7x}) = 7e^{7x}
\]
3. **Differentiate \( v \)** using the chain rule:
\[
v' = \frac{d}{dx}((7x + 3)^4) = 4(7x + 3)^3 \cdot 7 = 28(7x + 3)^3
\]
4. **Apply the product rule**:
\[
y' = u'v + uv' = (7e^{7x})(7x + 3)^4 + (e^{7x})(28(7x + 3)^3)
\]
5. **Combine the results**:
\[
y' = 7e^{7x}(7x + 3)^4 + 28e^{7x}(7x + 3)^3
\]
6. **Factor out common terms**:
\[
y' = e^{7x} \left[ 7(7x + 3)^4 + 28(7x + 3)^3 \right]
\]
This is the derivative of the given function.
To find the derivative of \( y = e^{7x} (7x + 3)^4 \), we will use the product rule and the chain rule.
Let ( u = e^{7x} ) and ( v = (7x + 3)^4 ).
Differentiate ( u ):
[
u' = 7e^{7x}
]
Differentiate ( v ) using the chain rule:
[
v' = 4(7x + 3)^3 \cdot 7 = 28(7x + 3)^3
]
Apply the product rule:
[
y' = u'v + uv'
]
Substituting ( u, u', v, v' ):
[
y' = (7e^{7x})(7x + 3)^4 + e^{7x}(28(7x + 3)^3)
]
Factor out common terms:
[
y' = e^{7x} \left( 7(7x + 3)^4 + 28(7x + 3)^3 \right)
]
So, the derivative is:
[
y' = e^{7x} \left( 7(7x + 3)^4 + 28(7x + 3)^3 \right)
]
example error 1
