differentiate the following function\n$y = sqrt{x + 5}(5x + 6)^2$\n$\frac{d}{dx}sqrt{x + 5}(5x + 6)^2=square$

Explanation:

Step1: Apply product - rule

The product - rule states that if $y = u\cdot v$, then $y^\prime=u^\prime v + uv^\prime$. Let $u=\sqrt{x + 5}=(x + 5)^{\frac{1}{2}}$ and $v=(5x + 6)^{2}$.

Step2: Differentiate $u$

Using the chain - rule, if $u=(x + 5)^{\frac{1}{2}}$, then $u^\prime=\frac{1}{2}(x + 5)^{-\frac{1}{2}}\cdot1=\frac{1}{2\sqrt{x + 5}}$.

Step3: Differentiate $v$

Using the chain - rule, if $v=(5x + 6)^{2}$, let $t = 5x+6$, then $v=t^{2}$. $\frac{dv}{dt}=2t$ and $\frac{dt}{dx}=5$. So $v^\prime=2(5x + 6)\cdot5 = 10(5x + 6)$.

Step4: Calculate $y^\prime$

$y^\prime=u^\prime v+uv^\prime=\frac{1}{2\sqrt{x + 5}}\cdot(5x + 6)^{2}+\sqrt{x + 5}\cdot10(5x + 6)$. Factor out $(5x + 6)$: $y^\prime=(5x + 6)\left(\frac{(5x + 6)}{2\sqrt{x + 5}}+10\sqrt{x + 5}\right)$. Get a common denominator: $y^\prime=(5x + 6)\left(\frac{5x + 6+20(x + 5)}{2\sqrt{x + 5}}\right)$. Expand the numerator: $y^\prime=(5x + 6)\left(\frac{5x + 6+20x+100}{2\sqrt{x + 5}}\right)=(5x + 6)\left(\frac{25x + 106}{2\sqrt{x + 5}}\right)=\frac{(5x + 6)(25x + 106)}{2\sqrt{x + 5}}$.

Answer:

$\frac{(5x + 6)(25x + 106)}{2\sqrt{x + 5}}$