differentiate. g(x)=(7x² + 5)(4x+√x) g(x)=

Explanation:

Step1: Apply product - rule

The product - rule states that if $G(x)=u(x)v(x)$, then $G^\prime(x)=u^\prime(x)v(x)+u(x)v^\prime(x)$. Let $u(x)=7x^{2}+5$ and $v(x)=4x + x^{\frac{1}{2}}$.

Step2: Differentiate $u(x)$

Differentiating $u(x)=7x^{2}+5$ with respect to $x$ using the power - rule $\frac{d}{dx}(ax^{n})=nax^{n - 1}$, we get $u^\prime(x)=\frac{d}{dx}(7x^{2}+5)=14x$.

Step3: Differentiate $v(x)$

Differentiating $v(x)=4x + x^{\frac{1}{2}}$ with respect to $x$ using the power - rule, we get $v^\prime(x)=\frac{d}{dx}(4x)+\frac{d}{dx}(x^{\frac{1}{2}})=4+\frac{1}{2}x^{-\frac{1}{2}}$.

Step4: Calculate $G^\prime(x)$

$G^\prime(x)=u^\prime(x)v(x)+u(x)v^\prime(x)$ $=14x(4x + x^{\frac{1}{2}})+(7x^{2}+5)(4+\frac{1}{2}x^{-\frac{1}{2}})$ $=56x^{2}+14x^{\frac{3}{2}}+28x^{2}+\frac{7}{2}x^{\frac{3}{2}}+20+\frac{5}{2}x^{-\frac{1}{2}}$ $=(56x^{2}+28x^{2})+(14x^{\frac{3}{2}}+\frac{7}{2}x^{\frac{3}{2}})+\frac{5}{2}x^{-\frac{1}{2}}+20$ $=84x^{2}+\frac{28x^{\frac{3}{2}} + 7x^{\frac{3}{2}}}{2}+\frac{5}{2\sqrt{x}}+20$ $=84x^{2}+\frac{35}{2}x^{\frac{3}{2}}+\frac{5}{2\sqrt{x}}+20$

Answer:

$84x^{2}+\frac{35}{2}x^{\frac{3}{2}}+\frac{5}{2\sqrt{x}}+20$