given ( y = f(u) ) and ( u = g(x) ), find ( \frac{dy}{dx}=f(g(x))g(x) ) for the following functions.\n(…

given ( y = f(u) ) and ( u = g(x) ), find ( \frac{dy}{dx}=f(g(x))g(x) ) for the following functions.\n( y=sin u, u = 4x - 3 )\n( \frac{dy}{dx}=f(g(x))g(x)=square )

given ( y = f(u) ) and ( u = g(x) ), find ( \frac{dy}{dx}=f(g(x))g(x) ) for the following functions.\n( y=sin u, u = 4x - 3 )\n( \frac{dy}{dx}=f(g(x))g(x)=square )

Answer

Explanation:

Step1: Find the derivative of (y = f(u)) with respect to (u)

Given (y=\sin u), by the derivative formula ((\sin x)^\prime=\cos x), we have (f^\prime(u)=\cos u). Then (f^\prime(g(x))=\cos(4x - 3)) (since (u = g(x)=4x-3)).

Step2: Find the derivative of (u = g(x)) with respect to (x)

Given (u = 4x-3), by the power rule ((ax + b)^\prime=a) ((a = 4), (b=-3)), we have (g^\prime(x)=4).

Step3: Apply the chain - rule formula (\frac{dy}{dx}=f^\prime(g(x))g^\prime(x))

Substitute (f^\prime(g(x))=\cos(4x - 3)) and (g^\prime(x)=4) into the formula (\frac{dy}{dx}=f^\prime(g(x))g^\prime(x)), we get (\frac{dy}{dx}=4\cos(4x - 3)).

Answer:

(4\cos(4x - 3))