To find the partial derivatives [tex]\( f_x \)[/tex] and [tex]\( f_y \)[/tex] for the function [tex]\( f(x, y) = 11(7x - 5y + 8)^9 \)[/tex], we need to differentiate the function with respect to [tex]\( x \)[/tex] and [tex]\( y \)[/tex] separately.
### Step-by-Step Solution:
1. Define the function:
[tex]\[
f(x, y) = 11(7x - 5y + 8)^9
\][/tex]
2. Partial Derivative with respect to [tex]\( x \)[/tex]:
Let's find [tex]\( f_x \)[/tex], the partial derivative of [tex]\( f \)[/tex] with respect to [tex]\( x \)[/tex].
- Applying the chain rule, differentiate the outer function and then the inner function.
[tex]\[
f_x = \frac{\partial}{\partial x} \left[ 11(7x - 5y + 8)^9 \right]
\][/tex]
- The outer function is [tex]\( 11u^9 \)[/tex], where [tex]\( u = 7x - 5y + 8 \)[/tex]. Differentiating [tex]\( 11u^9 \)[/tex] with respect to [tex]\( u \)[/tex] gives:
[tex]\[
99u^8 \quad (because \frac{\partial}{\partial u} 11u^9 = 99u^8)
\][/tex]
- Now, multiply by the derivative of the inner function [tex]\( 7x - 5y + 8 \)[/tex] with respect to [tex]\( x \)[/tex]:
[tex]\[
\frac{\partial}{\partial x} (7x - 5y + 8) = 7
\][/tex]
- Combining these results:
[tex]\[
f_x = 11 \cdot 9 \cdot (7x - 5y + 8)^8 \cdot 7 = 693(7x - 5y + 8)^8
\][/tex]
3. Partial Derivative with respect to [tex]\( y \)[/tex]:
Now, let's find [tex]\( f_y \)[/tex], the partial derivative of [tex]\( f \)[/tex] with respect to [tex]\( y \)[/tex].
- Again, applying the chain rule,
[tex]\[
f_y = \frac{\partial}{\partial y} \left[ 11(7x - 5y + 8)^9 \right]
\][/tex]
- As before, the outer function is [tex]\( 11u^9 \)[/tex], with [tex]\( u = 7x - 5y + 8 \)[/tex]. Differentiating [tex]\( 11u^9 \)[/tex] with respect to [tex]\( u \)[/tex] gives:
[tex]\[
99u^8
\][/tex]
- Now, multiply by the derivative of the inner function [tex]\( 7x - 5y + 8 \)[/tex] with respect to [tex]\( y \)[/tex]:
[tex]\[
\frac{\partial}{\partial y} (7x - 5y + 8) = -5
\][/tex]
- Combining these results:
[tex]\[
f_y = 11 \cdot 9 \cdot (7x - 5y + 8)^8 \cdot (-5) = -495(7x - 5y + 8)^8
\][/tex]
Therefore, the partial derivatives are:
[tex]\[
f_x(x, y) = 693(7x - 5y + 8)^8
\][/tex]
[tex]\[
f_y(x, y) = -495(7x - 5y + 8)^8
\][/tex]
