Find the equation of the tangent line to the curve :
`y=x^(3)-3x+5` at the point (2, 7)

Answer

To find the equation of the tangent line to the curve \( y = x^3 - 3x + 5 \) at the point \( (2, 7) \), we will follow these steps:

### Step 1: Differentiate the function
We start by differentiating the function \( y \) with respect to \( x \) to find the slope of the tangent line.

\[ \frac{dy}{dx} = \frac{d}{dx}(x^3 - 3x + 5) \]

Using the power rule and the constant rule, we differentiate each term:

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

### Step 2: Evaluate the derivative at the given point
Next, we need to find the slope of the tangent line at the point \( (2, 7) \). We substitute \( x = 2 \) into the derivative:

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

Calculating this gives:

\[ \frac{dy}{dx} \bigg|_{x=2} = 3(4) - 3 = 12 - 3 = 9 \]

So, the slope of the tangent line at the point \( (2, 7) \) is \( 9 \).

### Step 3: Use the point-slope form to find the equation of the tangent line
Now that we have the slope \( m = 9 \) and the point \( (x_1, y_1) = (2, 7) \), we can use the point-slope form of the equation of a line:

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

Substituting the values we have:

\[ y - 7 = 9(x - 2) \]

### Step 4: Simplify the equation
Now, we simplify the equation:

\[ y - 7 = 9x - 18 \]

Adding \( 7 \) to both sides gives:

\[ y = 9x - 11 \]

### Step 5: Rearranging to standard form
To write the equation in standard form, we can rearrange it:

\[ 9x - y - 11 = 0 \]

Thus, the equation of the tangent line is:

\[ 9x - y - 11 = 0 \]

### Summary
The equation of the tangent line to the curve \( y = x^3 - 3x + 5 \) at the point \( (2, 7) \) is:

\[ 9x - y - 11 = 0 \]