SLOPE OF A CURVE USING DERIVATIVE

Derivative measures the rate of change in one variable with respect to the change in other another variable.

For example, let x be an independent variable and y be a dependent variable which is depending on x.

So, the value of y is getting changed according to the change in the value of x.

Then, the derivative of y with respect to x is

dy/dx

The same dy/dx can also be defined as the slope of a curve at some value of x.

In finding slope of a curve, derivative measures the change in the values of y co-ordinate with respect to the change in the values of x co-ordinate. That is written as

dy/dx

Power Rule of Derivative

Power rule of derivative is the fundamental tool to find the derivative of a function y = xⁿ with respect to x, which is in the form

y = xⁿ

To get the derivative of xⁿ, we have to bring the exponent n in front of x and subtract 1 from the exponent.

dy/dx = nx^{n - 1}

Example 1 :

Find dy/dx, if y = x³.

Solution :

y = x³

dy/dx = 3x^{3 - 1}

= 3x²

Example 2 :

Find dy/dx, if y = x³.

Solution :

y = x²

dy/dx = 2x^{2 - 1}

= 2x¹

= 2x

Example 3 :

Find the slope of the curve y = 3x³ + 5 at x = -2.

Solution :

y = x³ + 5

Using the power rule of derivative,

dy/dx = 3x^{3 - 1} + 0

= 3x² + 0

= 3x²

Substitute x = -2.

= 3(-2)²

= 3(4)

= 12

Slope of the given curve at x = -2 is 12.

Example 4 :

Find the slope of the line y = 2x - 3 at x = 3.

Solution :

y = 2x - 3

Using the power rule of derivative,

dy/dx = 2(1) - 0

 = 2

Substitute x = 3.

= 2

In the derivative f'(x) = 2, if we substitute any value for x, we will be getting only 2 on the right side. Because there is no variable x on the right side.

From this, we can understand that the slope of a line is a constant. That is, for any value of x, the slope of a line will be same.

So, the slope of the given line is 2.

Example 5 :

Find the equation of a tangent to the curve y = x³ at x = 2.

Solution :

y = x³

Find dy/dx.

dy/dx = 3x^{3 - 1}

= 3x²

Substitute x = 2.

= 3(2)²

= 3(4)

= 12

Slope of the tangent at x = 2 is 12.

Equation of the tangent :

y = mx + b

Substitute m = 12.

y = 12x + b ----(1)

Substitute x = 2 in y = x³.

y = 2³

y = 8

x = 8 represents the point (2, 8).

Since the tangent is drawn to the curve at x = 2, the tangent is passing through the point (2, 8).

Substitute the point (2, 8) in (1).

8 = 12(2) + b

8 = 24 + b

Subtract 24 from both sides.

-16 = b

Substitute b = -16 in (1).

y = 12x - 16

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