Find the equation of the tangent line to the following curves :
`y=x^(3)" at "(1, 1)`

Answer

To find the equation of the tangent line to the curve \( y = x^3 \) at the point \( (1, 1) \), we can follow these steps:

### Step 1: Identify the point and the curve
We are given the curve \( y = x^3 \) and the point \( (1, 1) \).

### Step 2: Differentiate the curve to find the slope
To find the slope of the tangent line at the point \( (1, 1) \), we need to differentiate the curve with respect to \( x \).

\[ \frac{dy}{dx} = 3x^2 \]

### Step 3: Evaluate the derivative at \( x = 1 \)
Now, we substitute \( x = 1 \) into the derivative to find the slope at that point.

\[ \frac{dy}{dx} \bigg|_{x=1} = 3(1)^2 = 3 \]

### Step 4: Use the point-slope form of the equation of a line
The point-slope form of the equation of a line is given by:

\[ y - y_1 = m(x - x_1) \]

where \( (x_1, y_1) \) is the point on the line and \( m \) is the slope. Here, \( (x_1, y_1) = (1, 1) \) and \( m = 3 \).

Substituting these values into the equation:

\[ y - 1 = 3(x - 1) \]

### Step 5: Simplify the equation
Now, we simplify the equation:

\[ y - 1 = 3x - 3 \]

Adding 1 to both sides:

\[ y = 3x - 2 \]

### Step 6: Write the final equation
Thus, the equation of the tangent line to the curve \( y = x^3 \) at the point \( (1, 1) \) is:

\[ y = 3x - 2 \]

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