Find the derivative of y = 3 sin(4x^3 + 5) * sqrt(5(4x + 5/x)).

Step-by-step solution

1. 1

   Name the two factors so the product rule is visible. The function is a product, not a single composition, so start from $(uv)^{\prime}=u^{\prime}v+uv^{\prime}$ with $u=3\sin(4x^3+5),\qquad v=\sqrt{5\left(4x+\frac5x\right)}.$ Each factor is itself a composition, so each of the two derivatives will need the chain rule.

2. 2

   Differentiate the sine factor by the chain rule. With inner function $f(x)=4x^3+5$ we have $f^{\prime}(x)=12x^2$, so $u^{\prime}=3\cos(4x^3+5)\cdot 12x^2=36x^2\cos(4x^3+5).$

3. 3

   Differentiate the radical factor. Write $v=g(x)^{1/2}$ with $g(x)=5\left(4x+\frac5x\right)=20x+\frac{25}{x}$. Then $g^{\prime}(x)=20-\frac{25}{x^2}$, and since $\left(g^{1/2}\right)^{\prime}=\frac{g^{\prime}}{2\sqrt{g}}$, $v^{\prime}=\frac{20-\frac{25}{x^2}}{2\sqrt{5\left(4x+\frac5x\right)}}.$

4. 4

   Assemble with the product rule. Substituting the four pieces into $y^{\prime}=u^{\prime}v+uv^{\prime}$ gives $y^{\prime}=36x^2\cos(4x^3+5)\sqrt{5\left(4x+\frac5x\right)}+\frac{3\sin(4x^3+5)\left(20-\frac{25}{x^2}\right)}{2\sqrt{5\left(4x+\frac5x\right)}}.$

5. 5

   Note where the answer is valid. The term $5/x$ needs $x\ne 0$ and the square root needs $5\left(4x+\frac5x\right)\ge 0$, which holds for every $x>0$ but fails on part of the negative axis. Where the radicand is zero the radical sits in a denominator, so the derivative does not exist there either.