Question

27 - 76. calculate the derivative of the following functions. 27. $y=(3x^{2}+7x)^{10}$ 28. $y=(x^{2}+2x + 7)^{8}$ 29. $y=sqrt{10x + 1}$ 30. $y=sqrt3{x^{2}+9}$

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:

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