How to find the slope of a curve at a given point?

The slope of a curve is a slope of a tangent line for a curve at one point.

Answer: To find the slope of a curve at a given point, we simply differentiate the equation of the curve and find the first derivative of the curve, i.e., dy/dx.

Let's find out the answer with an example.

Explanation:

To determine slope, we use the following steps:

• Firstly, we need to differentiate the given equation or simply we have to find dy/dx of an equation.
• After differentiating we will get the equation of slope.
• Put the value of x in the equation to determine the slope.

Let's take an example to find the slope of a curve at a given point.

Example: Determine the slope of the curve y = x³ − x² + 1 at the given point (2,−15).

Given, 

Equation ⇒ y = x³ − x² + 1,

Point = (2,−15)

First we need to differentiate the given equation (y = x³ − x² + 1),

dy/dx = d(x³ − x² + 1)/dx

dy/dx = d(x³)/dx − d(x²)/dx + d(1)/dx

dy/dx = 3x² – 2x

Now, put the value of (x = 2) in the equation to determine the slope.

dy/dx = 3(2)² – 2(2)
dy/dx = 8 

Thus, 8 is the slope of the given curve.

Hence, to find the slope of a curve at a given point, we simply differentiate the equation of the curve and find the first derivative of the curve, i.e., dy/dx.