Show that $\frac{d}{dx} \left[ (2x)(3-5x)^{\frac{6}{5}} \right] = (6-22x)(3-5x)^{\frac{1}{5}}$. Hence, evaluate $\int x(3-5x)^{\frac{1}{5}} dx$.

1. Apply the product rule and chain rule to differentiate the expression.

Let $y = (2x)(3-5x)^{\frac{6}{5}}$.
Using the product rule $\frac{d}{dx}(uv) = u\frac{dv}{dx} + v\frac{du}{dx}$:
$\frac{dy}{dx} = (2x) \cdot \frac{d}{dx}\left[(3-5x)^{\frac{6}{5}}\right] + (3-5x)^{\frac{6}{5}} \cdot \frac{d}{dx}(2x)$
$\frac{dy}{dx} = (2x) \left[ \frac{6}{5}(3-5x)^{\frac{1}{5}}(-5) \right] + (3-5x)^{\frac{6}{5}}(2)$

2. Simplify the expression by factoring out the common term.

$\frac{dy}{dx} = -12x(3-5x)^{\frac{1}{5}} + 2(3-5x)^{\frac{6}{5}}$
Factor out $(3-5x)^{\frac{1}{5}}$:
$\frac{dy}{dx} = (3-5x)^{\frac{1}{5}} \left[ -12x + 2(3-5x)^{1} \right]$

3. Expand and simplify the terms inside the bracket to obtain the final result.

$\frac{dy}{dx} = (3-5x)^{\frac{1}{5}} \left[ -12x + 6 - 10x \right]$
$\frac{dy}{dx} = (3-5x)^{\frac{1}{5}} (6 - 22x)$
$\frac{dy}{dx} = (6-22x)(3-5x)^{\frac{1}{5}}$

1. Express the integral of the result from part (a) as the original function.

From part (a), we have:
$\int (6-22x)(3-5x)^{\frac{1}{5}} \, dx = 2x(3-5x)^{\frac{6}{5}} + C$

2. Rewrite the integrand to isolate the required integral.

Notice that $6-22x = 2(3-5x) - 12x$. Therefore:
$(6-22x)(3-5x)^{\frac{1}{5}} = [2(3-5x) - 12x](3-5x)^{\frac{1}{5}}$
$= 2(3-5x)^{\frac{6}{5}} - 12x(3-5x)^{\frac{1}{5}}$

Substituting this into the integral equation:
$\int \left[ 2(3-5x)^{\frac{6}{5}} - 12x(3-5x)^{\frac{1}{5}} \right] \, dx = 2x(3-5x)^{\frac{6}{5}} + C$
$\int 2(3-5x)^{\frac{6}{5}} \, dx - 12 \int x(3-5x)^{\frac{1}{5}} \, dx = 2x(3-5x)^{\frac{6}{5}} + C$

3. Evaluate the standard integral and solve for the required integral.

Evaluate $\int 2(3-5x)^{\frac{6}{5}} \, dx$:
$2 \left[ \frac{(3-5x)^{\frac{11}{5}}}{\frac{11}{5} \times (-5)} \right] = -\frac{2}{11}(3-5x)^{\frac{11}{5}}$

Substitute back:
$-\frac{2}{11}(3-5x)^{\frac{11}{5}} - 12 \int x(3-5x)^{\frac{1}{5}} \, dx = 2x(3-5x)^{\frac{6}{5}} + C$
$12 \int x(3-5x)^{\frac{1}{5}} \, dx = -\frac{2}{11}(3-5x)^{\frac{11}{5}} - 2x(3-5x)^{\frac{6}{5}} + C'$
$\int x(3-5x)^{\frac{1}{5}} \, dx = -\frac{1}{66}(3-5x)^{\frac{11}{5}} - \frac{1}{6}x(3-5x)^{\frac{6}{5}} + C''$

4. Factorise and simplify the final answer.

$\int x(3-5x)^{\frac{1}{5}} \, dx = (3-5x)^{\frac{6}{5}} \left[ -\frac{1}{66}(3-5x) - \frac{1}{6}x \right] + C$
$= (3-5x)^{\frac{6}{5}} \left[ -\frac{1}{22} + \frac{5}{66}x - \frac{11}{66}x \right] + C$
$= (3-5x)^{\frac{6}{5}} \left[ -\frac{1}{22} - \frac{6}{66}x \right] + C$
$= (3-5x)^{\frac{6}{5}} \left[ -\frac{1}{22} - \frac{1}{11}x \right] + C$
$= -\frac{1}{22}(1+2x)(3-5x)^{\frac{6}{5}} + C$