Differentiate y = 3x6. Find racdydx of y = 2x3 − 3x2 + 12x. Integrate integral (10x3 + 5x + 3x)\,dx. Construct the following angles.

Using the power rule for differentiation, $\frac{d}{dx}(ax^n) = anx^{n-1}$:

$$\frac{dy}{dx} = 3 \times 6 x^{6-1}$$ $$\frac{dy}{dx} = 18x^5$$

The derivative is $\boxed{18x^5}$.

Differentiate each term using the power rule:

$$\frac{dy}{dx} = \frac{d}{dx}(2x^3) - \frac{d}{dx}(3x^2) + \frac{d}{dx}(12x)$$ $$\frac{dy}{dx} = (2 \times 3 x^{3-1}) - (3 \times 2 x^{2-1}) + (12 \times 1 x^{1-1})$$ $$\frac{dy}{dx} = 6x^2 - 6x + 12x^0$$ $$\frac{dy}{dx} = 6x^2 - 6x + 12$$

The derivative is $\boxed{6x^2 - 6x + 12}$.

First, simplify the integrand:

$$\int (10x^3 + 8x)\,dx$$

Using the power rule for integration, $\int ax^n\,dx = a\frac{x^{n+1}}{n+1} + C$:

$$\int 10x^3\,dx + \int 8x\,dx$$ $$10\frac{x^{3+1}}{3+1} + 8\frac{x^{1+1}}{1+1} + C$$ $$10\frac{x^4}{4} + 8\frac{x^2}{2} + C$$ $$\frac{5}{2}x^4 + 4x^2 + C$$

The integral is $\boxed{\frac{5}{2}x^4 + 4x^2 + C}$.

These steps describe how to construct the angles using a compass and straightedge.

i) $90^\circ$

• Draw a line segment and mark a point P on it.
• With P as the center, draw an arc intersecting the line at two points, A and B.
• With A and B as centers, and a radius greater than AP, draw two arcs intersecting each other at point C.
• Join P and C. The angle $\angle CPA$ (or $\angle CPB$) is $90^\circ$.

ii) $45^\circ$

• Construct a $90^\circ$ angle, say $\angle CPB$, as described above.
• Bisect the $90^\circ$ angle. With B and C as centers, and a radius greater than half of BC, draw two arcs intersecting each other at point D.
• Join P and D. The angle $\angle DPB$ is $45^\circ$.

iii) $60^\circ$

• Draw a line segment and mark a point P on it.
• With P as the center, draw an arc intersecting the line at point A.
• With A as the center and the same radius, draw another arc intersecting the first arc at point B.
• Join P and B. The angle $\angle BPA$ is $60^\circ$.

iv) $120^\circ$

• Draw a line segment and mark a point P on it.
• With P as the center, draw an arc intersecting the line at point A.
• With A as the center and the same radius, draw an arc intersecting the first arc at point B (this forms $60^\circ$).
• With B as the center and the same radius, draw another arc intersecting the first arc at point C.
• Join P and C. The angle $\angle CPA$ is $120^\circ$.

v) $150^\circ$

• Draw a line segment AB and mark a point P on it.
• With P as the center, draw an arc intersecting AB at points X and Y.
• With X as the center and the same radius, draw an arc intersecting the first arc at Z. This forms $\angle ZPX = 60^\circ$.
• Bisect $\angle ZPX$. With X and Z as centers, draw arcs intersecting at M. Join P and M. This forms $\angle MPX = 30^\circ$.
• The angle $\angle MPY$ is $180^\circ - 30^\circ = 150^\circ$.