If y = $\sqrt { u }$ and u = $x ^ { 3 }$

Question

If y = $\sqrt { u }$ and u = $x ^ { 3 }$ - 5 $x ^ { 2 }$ + 1, find $\frac { d y } { d x }$ .
A) $\sqrt { 3 x ^ { 2 } - 10 x }$
B) $\sqrt { 3 x ^ { 2 } - 10 x }$ $\left( x ^ { 3 } \right.$ - $5 x ^ { 2 }$ + 1)
C) $\frac { 3 x ^ { 2 } - 10 x } { 2 \sqrt { x ^ { 3 } - 5 x ^ { 2 } + 1 } }$
D) $\sqrt { x ^ { 3 } - 5 x ^ { 2 } + 1 }$

Quizplus · Accepted Answer

To find $\frac{dy}{dx}$, we use the chain rule. First, differentiate $y = \sqrt{u}$ with respect to $u$ to get $\frac{1}{2\sqrt{u}}$, then differentiate $u = x^3 - 5x^2 + 1$ with respect to $x$ to get $3x^2 - 10x$. Multiply these derivatives together to get $\frac{3x^2 - 10x}{2\sqrt{x^3 - 5x^2 + 1}}$.

If Y = u\sqrt { u }u​ And U = x3x ^ { 3 }x3