Question

Find the derivative of y = 2(22x+31)^(3/2) + arctan((22x+21)^(1/2)) + ln(11x+6) - 22x^2.

Answer 1

To find the derivative of the function

\[
y = 2(22x+31)^{3/2} + \arctan((22x+21)^{1/2}) + \ln(11x+6) - 22x^2,
\]

we will take the derivative of each term separately.

1. **Derivative of \(2(22x+31)^{3/2}\)**:

Using the chain rule:

\[
\frac{d}{dx}[2(22x+31)^{3/2}] = 2 \cdot \frac{3}{2}(22x+31)^{1/2} \cdot (22) = 3(22)(22x + 31)^{1/2} = 66(22x + 31)^{1/2}.
\]

2. **Derivative of \(\arctan((22x+21)^{1/2})\)**:

Using the chain rule again:

\[
\frac{d}{dx}[\arctan((22x+21)^{1/2})] = \frac{1}{1 + ((22x+21)^{1/2})^2} \cdot \frac{1}{2}(22x + 21)^{-1/2} \cdot (22) = \frac{22}{1 + (22x + 21)} \cdot \frac{1}{2(22x + 21)^{1/2}}.
\]

This simplifies to:

\[
\frac{11}{(22x + 21) + 1}\cdot \frac{1}{(22x + 21)^{1/2}} = \frac{11}{23 + 22x} \cdot \frac{1}{(22x + 21)^{1/2}}.
\]

3. **Derivative of \(\ln(11x + 6)\)**:

Using the derivative of the natural logarithm:

\[
\frac{d}{dx}[\ln(11x + 6)] = \frac{11}{11x + 6}.
\]

4. **Derivative of \(-22x^2\)**:

This is simply:

\[
\frac{d}{dx}[-22x^2] = -44x.
\]

Now, combining all of these results, we get:

\[
y' = 66(22x + 31)^{1/2} + \frac{11}{(22x + 22)^{1/2}(23 + 22x)} + \frac{11}{11x + 6} - 44x.
\]

Thus, the complete derivative of \(y\) is:

\[
y' = 66(22x + 31)^{1/2} + \frac{11}{(22x + 21)^{1/2}(23 + 22x)} + \frac{11}{11x + 6} - 44x.
\]