27 - 76. calculate the derivative of the following functions. 27. $y=(3x^{2}+7x)^{10}$ 28. $y=(x^{2}+2x +…

Explanation:

Step1: Identify the outer - function and inner - function for the chain rule

For (y=(3x^{2}+7x)^{10}), let (u = 3x^{2}+7x), then (y = u^{10}). The chain - rule states that (\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}).

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

Differentiate (y = u^{10}) with respect to (u). Using the power rule (\frac{d}{du}(u^{n})=nu^{n - 1}), we get (\frac{dy}{du}=10u^{9}).

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

Differentiate (u = 3x^{2}+7x) with respect to (x). Using the power rule (\frac{d}{dx}(ax^{n})=nax^{n - 1}), we have (\frac{du}{dx}=6x + 7).

Step4: Apply the chain - rule

Substitute (u = 3x^{2}+7x) back into (\frac{dy}{du}) and then multiply by (\frac{du}{dx}). So (\frac{dy}{dx}=10(3x^{2}+7x)^{9}(6x + 7)).

Answer:

(10(3x^{2}+7x)^{9}(6x + 7))