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To find the equation of the tangent to the curve \( y = x^3 - 3x^2 + 4x - 5 \) at the point \( (1, -3) \), we need to follow these steps:

### Step 1: Differentiate the Curve Equation
The equation of the curve is:
\[
y = x^3 - 3x^2 + 4x - 5
\]

To find the slope of the tangent line, we need the first derivative \( y' \) of \( y \) with respect to \( x \):
\[
y' = \frac{d}{dx}(x^3 - 3x^2 + 4x - 5)
\]

Let's differentiate term by term:
\[
y' = 3x^2 - 6x + 4
\]

### Step 2: Evaluate the Derivative at the Given Point
Now, we evaluate \( y' \) at \( x = 1 \) to find the slope of the tangent at \( (1, -3) \):
\[
y'|_{x=1} = 3(1)^2 - 6(1) + 4 = 3 - 6 + 4 = 1
\]

So, the slope of the tangent line at \( (1, -3) \) is \( m = 1 \).

### Step 3: Use the Point-Slope Form to Find the Equation of the Tangent Line
The point-slope form of a line is given by:
\[
y - y_1 = m(x - x_1)
\]

Substitute \( m = 1 \), \( x_1 = 1 \), and \( y_1 = -3 \):
\[
y + 3 = 1(x - 1)
\]

Simplify this equation:
\[
y + 3 = x - 1
\]
\[
y = x - 4
\]

### Answer
The equation of the tangent to the curve at the point \( (1, -3) \) is:
\[
\boxed{y = x - 4}
\]

Would you like further details or have any questions?

Here are some related questions you might find interesting:
1. How do you find the equation of a normal line to a curve at a given point?
2. What is the significance of the derivative in determining the slope of a tangent?
3. How do you find the second derivative of a function, and what does it represent?
4. How can you determine if a point is a local maximum or minimum using the derivative?
5. How can you use implicit differentiation to find the tangent to curves defined implicitly?

**Tip:** Remember that the slope of the tangent line to a curve at a point gives the instantaneous rate of change of the curve at that point.

Find the equation of the tangent to the curve y = x^3 - 3x^2 + 4x - 5 at the point (1, -3).

Learn how to find the equation of the tangent to the cubic curve y = x^3 - 3x^2 + 4x - 5 at the point (1, -3). This solution involves finding the derivative of the function and using the point-slope form of a line. Suitable for students in Grades 10-12.

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