To find the derivative of the given function [tex]\( y = (1 + x^2)^{19} \)[/tex] with respect to [tex]\( x \)[/tex], we can follow these steps:
1. Identify the function: The given function is:
[tex]\[
y = (1 + x^2)^{19}
\][/tex]
2. Use the chain rule: To differentiate [tex]\( y \)[/tex] with respect to [tex]\( x \)[/tex], we will apply the chain rule. The chain rule states that if you have a composite function [tex]\( y = f(g(x)) \)[/tex], then the derivative [tex]\( \frac{d y}{d x} \)[/tex] is given by:
[tex]\[
\frac{d y}{d x} = \frac{d f}{d g} \cdot \frac{d g}{d x}
\][/tex]
Here, [tex]\( f(u) = u^{19} \)[/tex] where [tex]\( u = 1 + x^2 \)[/tex].
3. Differentiate the outer function: First, we differentiate the outer function [tex]\( f(u) = u^{19} \)[/tex] with respect to [tex]\( u \)[/tex]:
[tex]\[
\frac{d f}{d u} = 19 u^{18}
\][/tex]
Since [tex]\( u = 1 + x^2 \)[/tex], this becomes:
[tex]\[
\frac{d f}{d u} = 19 (1 + x^2)^{18}
\][/tex]
4. Differentiate the inner function: Next, we differentiate the inner function [tex]\( g(x) = 1 + x^2 \)[/tex] with respect to [tex]\( x \)[/tex]:
[tex]\[
\frac{d g}{d x} = 2x
\][/tex]
5. Multiply the derivatives: Now, we multiply the derivatives obtained from the outer and inner functions:
[tex]\[
\frac{d y}{d x} = \left( 19 (1 + x^2)^{18} \right) \cdot \left( 2x \right)
\][/tex]
6. Simplify the expression: Combine and simplify the expression:
[tex]\[
\frac{d y}{d x} = 38 x (1 + x^2)^{18}
\][/tex]
7. Relate to [tex]\( y \)[/tex]: Since [tex]\( y = (1 + x^2)^{19} \)[/tex], we notice that:
[tex]\[
(1 + x^2)^{18} = \frac{y}{1 + x^2}
\][/tex]
Therefore,
[tex]\[
\frac{d y}{d x} = 38 x \left( \frac{y}{1 + x^2} \right)
\][/tex]
Simplify this to:
[tex]\[
\frac{d y}{d x} = \frac{38 x y}{1 + x^2}
\][/tex]
8. Select the correct answer: Comparing this result with the provided options, we see that the correct option is:
[tex]\[
\boxed{\text{B}}
\][/tex]
