FIND THE GRADIENT OF THE TANGENT LINE WITH A GIVEN POINT

Find the gradient of the tangent to the curve at the point indicated :

Problem 1 :

y = x² + 3x  (2, 10)

Solution :

y = x² + 3x

dy/dx = 2x + 3

Put x = 2,

dy/dx = 2(2) + 3

= 4 + 3

dy/dx = 7

Problem 2 :

y = 2x³ - 4   (3, 50)

Solution :

y = 2x³ - 4

dy/dx = 2(3x²) - 0

dy/dx = 6x²

Put x = 3,

dy/dx = 6(3)²

dy/dx = 54

Problem 3 :

y = -x² + 1/x  (-2, -4.5)

Solution :

y = -x² + 1/x 

y = -x² + (x^-1)

dy/dx = -2x - x^(-1-1)

= -2x - x^-2

dy/dx = -2x - 1/x²

Put x = -2,

dy/dx = -2(-2) - 1/(-2)²

= 4 - 1/4

dy/dx = 15/4

Find the equation of the tangent to the curve at the point indicated:

Problem 4 :

y = 3x² - x  (1, 2)

Solution :

y = 3x² - x

dy/dx = 6x - 1

Put x = 1,

dy/dx = 6(1) - 1

dy/dx = 5

Equation of tangent:

y - y₁ = m(x - x₁)

y - 2 = 5(x - 1)

y - 2 = 5x - 5

y = 5x - 5 + 2

y = 5x - 3

Problem 5 :

y = x³ + 4x  (-1, -5)

Solution :

y = x³ + 4x 

dy/dx = 3x² + 4

Put x = 1,

dy/dx = 3(1)² + 4

dy/dx = 7

Equation of tangent:

y - y₁ = m(x - x₁)

y + 5 = 7(x + 1)

y + 5 = 7x + 7

y = 7x + 7 - 5

y = 7x + 2

Problem 6 :

y = x² - 1/x  (1, 2)

Solution :

y = x² - 1/x

y= x² - (x^-1)

dy/dx = 2x - (-x^-2)

dy/dx = 2x + 1/x²

Put x = 1,

dy/dx = 2(1) + 1/(1)²

dy/dx = 3

Equation of tangent:

y - y₁ = m(x - x₁)

y - 2 = 3(x - 1)

y - 2 = 3x - 3

y = 3x - 3 + 2

y = 3x - 1