If $y = (5 x + 1)^{4}$ , find $\frac{d y}{d x}$ when $x = 1$ .

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The curve $y = x^{3} + 3 x^{2} - 9 x + 7$ has two stationary points. Find the difference between their $y$ -coordinates (larger minus smaller).
Find the volume of the solid formed by rotating $y = 3 x$ about the $x$ -axis from $x = 0$ to $x = 1$ , in terms of $π$ .
Evaluate $x \to 0 lim \frac{sin 6 x}{2 x}$ .
If $y = ln (4 x + 5)$ , find $\frac{d y}{d x}$ at $x = 1$ .
Find the local minimum value of $y = x^{3} + 6 x^{2} + 9 x + 7$ .
Find the area under the curve $y = 5 x^{2}$ between $x = 0$ and $x = 3$ .