please answer them.. class 11th differentiation problems..
1. Find d/dt. ( 4t²- t) ( t³- 8t²+ 12)
2. Find d/dx Sinx.x-² ( this is sinx. x ki power-2)
3.Find d/dx of x³.e^x ( this is x³.e ki power x)
4. y = (4x+9) (8x+1)
5. y= x²/ cosx
6. y= 1/x
7. y= e⁴^x/ cosx ( this is eki power 4x/ cosx)
8.y= sinx.x³+ cosx.x²
please someone help, guys.. please, it's urgent.. please.... it will be ok with answers only, but if someone will explain it will more helpful.. its a request.. please help
Answers
Answer:
1.
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)