If f(x) = x $( x - 1 ) ^ { 5 }$ and g(x) = $x ^ { 2 }$ , find $\frac { \mathrm { d } } { \mathrm { dx } }$ f(g(x)).
A) 2x $\left( x ^ { 2 } - 1 \right) ^ { 5 }$ + 10 $x ^ { 2 }$ $\left( x ^ { 2 } - 1 \right) ^ { 4 }$
B) $\left( x ^ { 2 } - 1 \right) ^ { 5 }$ + 5 $x ^ { 2 }$ $\left( x ^ { 2 } - 1 \right) ^ { 4 }$
C) 2x $\left( x ^ { 2 } - 1 \right) ^ { 5 }$ + 10 $x ^ { 3 }$ ( $x ^ { 2 }$ - 1)
D) 2x $\left( x ^ { 2 } - 1 \right) ^ { 5 }$ + 10 $x ^ { 3 }$ $\left( x ^ { 2 } - 1 \right) ^ { 4 }$

Question

Quizplus · Accepted Answer

To find $\frac { \mathrm { d } } { \mathrm { dx } }$ f(g(x)), we first substitute g(x) into f(x), getting f(g(x)) = $x^2((x^2 - 1)^5)$. Then, applying the chain rule and product rule, we differentiate this expression with respect to x, leading to the correct answer.

If F(x) = X (x−1)5( x - 1 ) ^ { 5 }(x−1)5

If F(x) = X $( x - 1 ) ^ { 5 }$