Question

suppose ( y=sin (5 x^{2}+2 x - 5) ). we can write ( y=sin (u) ), where ( u=square ). the leibnitz notation for the chain rule is ( \frac{d y}{d x}=\frac{d y}{d u} cdot \frac{d u}{d x} ). the factors are ( \frac{d y}{d u}=square ) (written as a function of ( u ) ) and ( \frac{d u}{d x}=square ). now substitute in the function of ( x ) for ( u ) to get ( \frac{d y}{d x}=square ) (written as a function of ( x ) ).

Explanation:

Step1: Find \(u\)

Given \(y = \sin(5x^{2}+2x - 5)\), if \(y=\sin(u)\), then \(u = 5x^{2}+2x - 5\).

Step2: Find \(\frac{dy}{du}\)

Since \(y=\sin(u)\), using the derivative formula \(\frac{d}{du}(\sin(u))=\cos(u)\), we have \(\frac{dy}{du}=\cos(u)\).

Step3: Find \(\frac{du}{dx}\)

Since \(u = 5x^{2}+2x - 5\), using the power rule \(\frac{d}{dx}(ax^{n})=nax^{n - 1}\), we get \(\frac{du}{dx}=\frac{d}{dx}(5x^{2}+2x - 5)=10x + 2\).

Step4: Find \(\frac{dy}{dx}\)

By the chain - rule \(\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}\). Substitute \(\frac{dy}{du}=\cos(u)\) and \(\frac{du}{dx}=10x + 2\) into the formula. Then replace \(u\) with \(5x^{2}+2x - 5\), so \(\frac{dy}{dx}=\cos(5x^{2}+2x - 5)\cdot(10x + 2)\).

Answer:

\(u = 5x^{2}+2x - 5\); \(\frac{dy}{du}=\cos(u)\); \(\frac{du}{dx}=10x + 2\); \(\frac{dy}{dx}=(10x + 2)\cos(5x^{2}+2x - 5)\)