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.
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).
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.