FINDING THE SLOPE OF A TANGENT LINE USING DERIVATIVE

You follow the steps given below to find the slope of a tangent line to a curve at a given point using derivative.

Step 1 :

Let y = f(x) be the function which represents a curve. Find the derivative ᵈʸ⁄dor f'(x), where ᵈʸ⁄d  or f'(x)  is the slope of the line tangent to the curve at any point.

Step 2 :

Substiute the given point into ᵈʸ⁄dor f'(x) and evaluate.

In each case, find the slope of a line tagent to the graph of the function at the given point.

Example 1 :

f(x) = 3 - 5x,  (-1, 8)

Solution :

f(x) = 3 - 5x

f'(x) = -5(1)

f'(x) -5

Slope of the tangent line at (-1, 8) :

m = 5

Example 2 :

Solution :

Slope of the tangent line at (-2, -2) :

Example 3 :

f(x) = 2x2 - 3,  (2, 5)

Solution :

f(x) = 2x2 - 3

f'(x) = -2(2x) - 0

f'(x) = -4x

Slope of the tangent line at (2, 5) :

m = -4(2)

m = -8

Example 4 :

f(x) = 5 - x2,  (3, -4)

Solution :

f(x) = 5 - x2

f'(x) = 0 - 2x

f'(x) = 2x

Slope of the tangent line at (3, -4) :

m = 2(3)

m = 6

Example 5 :

f(t) = 3t - t2,  (0, 0)

Solution :

f(t) = 3t - t2

f'(t) = 3(1) - 2t

f'(t) = 3 - 2t

Slope of the tangent line at (0, 0) :

m = 3 - 2(0)

m = 3

Example 6 :

h(t) = t2 + 4t,  (1, 5)

Solution :

h(t) = t2 + 4t

h'(t) = 2t - 4(1)

h'(t) = 2t - 4

Slope of the tangent line at (1, 5) :

m = 2(5) - 4

m = 10 - 4

m = 6

Example 7 :

x2x2 = 25,  (6, 3)

Solution :

x2 + x2 = 25

Find the derivative on both sides with respect to x.

Slope of the tangent line at (6, 3) :

Example 8 :

x = 3t  and  y = t2 + 1,  t = 2.

Solution :

Slope of the tangent line at t = 2 :

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