Question

f(t)=(4t*2-t)(t*3-8t*2+12) find the derivation of the given function​

Answer 1

Solution :

By solving the given equation, we get :

⠀⠀=> (4t² - t)(t³ - 8t² + 12)

⠀⠀=> 4t²(t³ - 8t² + 12) - t(t³ - 8t² + 12)

⠀⠀=> 4t⁶ - 32t⁴ + 48t² - t⁴ - 8t³ + 12t

⠀⠀=> 4t⁶ - 31t⁴ + 48t² - 8t³ + 12t

⠀⠀⠀⠀⠀∴ 4t⁶ - 31t⁴ + 48t² - 8t³ + 12t

Now by differentiating the equation w.r.t. t, we get :

⠀⠀=> dy/dy = d(4t⁶ - 31t⁴ + 48t² - 8t³ + 12t)/dt

⠀⠀=> dy/dt = d(4t⁶)/dx - d(31t⁴)/dx + d(48t²)/dx - d(8t³)/dx + d(12t)/dt

Now, let's find out the derivative of each term in the equation.

• Derivative of 4t⁶ :

By applying the power rule of differentiation [d(x^n)/dx = nx^(n - 1)], we get :

⠀⠀=> dy/dt = d(4t⁶)/dt

⠀⠀=> dy/dt = 6 × 4t⁽⁶ ⁻ ¹⁾

⠀⠀=> dy/dt = 6 × 4t⁵

⠀⠀=> dy/dt = 24t⁵

⠀⠀⠀ ∴ d(4t⁶)/dt = 24t⁵

Hence the derivative of 4t⁶ is 24t⁵.

• Differentiation of 31t⁴ :

By applying the power rule of differentiation [d(x^n)/dx = nx^(n - 1)], we get :

⠀⠀=> dy/dt = d(31t⁴)/dt

⠀⠀=> dy/dt = 4 × 31t⁽⁴ ⁻ ¹⁾

⠀⠀=> dy/dt = 4 × 31t³

⠀⠀=> dy/dt = 124³

⠀⠀⠀ ∴ d(31t⁴)/dt = 124t³

Hence the derivative of 31t⁴ is 124t³.

• Differentiation of 48t² :

By applying the power rule of differentiation [d(x^n)/dx = nx^(n - 1)], we get :

⠀⠀=> dy/dt = d(48t²)/dt

⠀⠀=> dy/dt = 2 × 48t⁽² ⁻ ¹⁾

⠀⠀=> dy/dt = 2 × 48t⁵

⠀⠀=> dy/dt = 96t

⠀⠀⠀ ∴ d(48t²)/dt = 96t

Hence the derivative of 48t² is 96t.

• Differentiation of 8t³ :

By applying the power rule of differentiation [d(x^n)/dx = nx^(n - 1)], we get :

⠀⠀=> dy/dt = d(8t³)/dt

⠀⠀=> dy/dt = 3 × 8t⁽³ ⁻ ¹⁾

⠀⠀=> dy/dt = 3 × 8t²

⠀⠀=> dy/dt = 24t²

⠀⠀⠀ ∴ d(8t³)/dt = 24t²

Hence the derivative of 8t³ is 24t².

• Differentiation of 12t :

By applying the power rule of differentiation [d(x^n)/dx = nx^(n - 1)], we get :

⠀⠀=> dy/dt = d(12t)/dt

⠀⠀=> dy/dt = 1 × 4t⁽¹ ⁻ ¹⁾

⠀⠀=> dy/dt = 1 × 12t⁰

⠀⠀=> dy/dt = 12

⠀⠀⠀ ∴ d(12t)/dt = 12

Hence the derivative of 12t is 12.

Now by substituting the derivate of the functions, we get :

⠀⠀=> dy/dt = d(4t⁶)/dx - d(31t⁴)/dx + d(4t²)/dx - d(8t³)/dx + d(12t)/dt

⠀⠀=> dy/dt = 24t⁵ - 124t³ + 96t - 23t² + 12

Thus the derivative of (4t² - t)(t³- 8t × 2 + 12) is (24t⁵ - 124t³ + 8t - 24t² + 12).