Find the slope of tengents drawn of the following curves at the given points:
(i) Curve ` y =x^(3)+1` at point (0, 1)
(ii) Curve ` x^(2)-y^(2)` = 20 at point (6, 4)
(iii) Curve ` y^(2)=4x` at point (1, 2)
(iv) Curve` y^(2) = 4" ax at point "(a/m^(2),(2a)/m)`

Answer

To find the slope of the tangents drawn to the given curves at the specified points, we will follow these steps for each curve:

### (i) Curve: \( y = x^3 + 1 \) at point \( (0, 1) \)

1. **Differentiate the function**:
\[ \frac{dy}{dx} = \frac{d}{dx}(x^3 + 1) = 3x^2 \]

2. **Evaluate the derivative at the given point** \( (0, 1) \):
\[ \frac{dy}{dx} \bigg|_{x=0} = 3(0)^2 = 0 \]

3. **Conclusion**: The slope of the tangent at the point \( (0, 1) \) is \( 0 \).

### (ii) Curve: \( x^2 - y^2 = 20 \) at point \( (6, 4) \)

1. **Differentiate implicitly**:
\[ 2x - 2y \frac{dy}{dx} = 0 \implies \frac{dy}{dx} = \frac{x}{y} \]

2. **Evaluate the derivative at the given point** \( (6, 4) \):
\[ \frac{dy}{dx} \bigg|_{(6, 4)} = \frac{6}{4} = \frac{3}{2} \]

3. **Conclusion**: The slope of the tangent at the point \( (6, 4) \) is \( \frac{3}{2} \).

### (iii) Curve: \( y^2 = 4x \) at point \( (1, 2) \)

1. **Differentiate implicitly**:
\[ 2y \frac{dy}{dx} = 4 \implies \frac{dy}{dx} = \frac{4}{2y} = \frac{2}{y} \]

2. **Evaluate the derivative at the given point** \( (1, 2) \):
\[ \frac{dy}{dx} \bigg|_{(1, 2)} = \frac{2}{2} = 1 \]

3. **Conclusion**: The slope of the tangent at the point \( (1, 2) \) is \( 1 \).

### (iv) Curve: \( y^2 = 4ax \) at point \( \left(\frac{a}{m^2}, \frac{2a}{m}\right) \)

1. **Differentiate implicitly**:
\[ 2y \frac{dy}{dx} = 4a \implies \frac{dy}{dx} = \frac{4a}{2y} = \frac{2a}{y} \]

2. **Evaluate the derivative at the given point** \( \left(\frac{a}{m^2}, \frac{2a}{m}\right) \):
\[ \frac{dy}{dx} \bigg|_{\left(\frac{a}{m^2}, \frac{2a}{m}\right)} = \frac{2a}{\frac{2a}{m}} = m \]

3. **Conclusion**: The slope of the tangent at the point \( \left(\frac{a}{m^2}, \frac{2a}{m}\right) \) is \( m \).

### Summary of Results:
- (i) Slope at \( (0, 1) \): \( 0 \)
- (ii) Slope at \( (6, 4) \): \( \frac{3}{2} \)
- (iii) Slope at \( (1, 2) \): \( 1 \)
- (iv) Slope at \( \left(\frac{a}{m^2}, \frac{2a}{m}\right) \): \( m \)