Given y = f(u) and u = g(x), find dy/dx = f(g(x))g(x).
-$$y = \csc u , u = x ^ { 7 } + 10 x$$
A) $- \left( 7 x ^ { 6 } + 10 \right) \cot ^ { 2 } \left( x ^ { 7 } + 10 x \right)$
B) $- \left( 7 x ^ { 6 } + 10 \right) \csc x \cot x$
C) $- \csc \left( x ^ { 7 } + 10 x \right) \cot \left( x ^ { 7 } + 10 x \right)$
D) $- \left( 7 x ^ { 6 } + 10 \right) \csc \left( x ^ { 7 } + 10 x \right) \cot \left( x ^ { 7 } + 10 x \right)$

Question

Quizplus · Accepted Answer

Using the chain rule, we have:

$$ \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} = \frac{d}{du}\left(\csc u\right) \cdot \frac{d}{dx}\left(x^7 + 10x\right) = -\csc u \cot u \cdot (7x^6+10)$$

Substituting back in for u, we get:

$$ -\csc\left(x^7+10x\right) \cot\left(x^7+10x\right) \cdot (7x^6+10) $$

So the correct answer is D.

Given Y = F(u) and U = G(x), Find Dy/dx y=csc⁡u,u=x7+10xy = \csc u , u = x ^ { 7 } + 10 xy=cscu,u=x7+10x

Given Y = F(u) and U = G(x), Find Dy/dx $y = \csc u , u = x ^ { 7 } + 10 x$