Differentiation and Integration of Sums of Multiples of Powers of x

Differentiation and Integration of Sums of Multiples of Powers of x

Differentiation

Differentiation is a process in calculus that gives us a way to calculate the rate at which a function is changing at any point.

  • The Power Rule states that if f(x) = xn, then f’(x) = nxn-1.

  • For any constant c, the derivative of c with respect to x is zero i.e. if f(x) = c, then f’(x) = 0.

  • The derivative of a constant multiplied by a function is the constant multiplied by the derivative of the function i.e. if f(x) = cu(x), then f’(x) = cu’(x).

  • The Sum Rule states that the derivative of a sum of functions is the sum of their derivatives i.e. if f(x) = u(x) + v(x), then f’(x) = u’(x) + v’(x).

Integration

Integration is the reverse process of differentiation. It’s used to calculate the area under a curve, or to find a function when given its derivative.

  • The Power Rule for Integration states that if f(x) = xn, then ∫f(x) dx = (1/(n+1)) xn+1 + c.

  • The integral of a constant c with respect to x is the constant multiplied by x i.e. if f(x) = c, then ∫f(x) dx = cx + c.

  • The integral of a constant multiplied by a function is the constant multiplied by the integral of the function i.e. if f(x) = cu(x), then ∫f(x) dx = c ∫u(x) dx.

  • The Sum Rule for Integration states that the integral of a sum of functions is the sum of their integrals i.e. if f(x) = u(x) + v(x), then ∫f(x) dx = ∫u(x) dx + ∫v(x) dx.

General tips

  • Always remember to add the constant of integration ‘c’ when integrating a function.

  • Make sure to correctly apply the power rule during differentiation and integration.

  • Use the sum rule to break down complex functions into simpler functions that can be integrated or differentiated individually.

  • When dealing with sums or differences of multiples of powers of x, deal with each term individually.

  • Also remember that the powers of x applies to each term within a sum separately when using the sum rule in both differentiation and integration.

Remember, practise makes perfect; work out several problems on differentiation and integration to become comfortable with these concepts. Keep these key points in mind and you’ll be able to tackle any problem on differentiation and integration of sums of multiples of powers of x.