Differentiation of Algebraic Functions II

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In today’s class, we will be talking more about differentiation of algebraic functions. Enjoy the class!

Differentiation of algebraic functions II

Differentiation of algebraic functions | classnotes.ng

Rules of differentiation

The basic rules of differentiation are presented here along with several examples. Remember that if y = f(x) is a function then the derivative of y can be represented by dy/dx or y′ or f′ or df /dx.

  • Rule 1:

The Derivative of a Constant. The derivative of a constant is zero.

  • Rule 2:

The General Power Rule. The derivative of xn is nxn−1.

  • Rule 3:

The Derivative of a Constant times a Function.

The derivative of kf (x), where k is a constant, is kf′ (x).

For example:

Differentiate y = 3x2.

In this case f(x) = x2 and k = 3, therefore the derivative is 3×2x1 = 6x.

  • Rule 4:

The Derivative of a Sum or a Difference.

If f(x) = h(x)±g(x), then df/dx = dh/dx ± dg/dx.

Example:

Differentiate f(x) = 3x2 −7x.

In this case k(x) = 3x2 and g(x) = 7x and so dk/dx = 6x and dg/dx = 7.

Therefore, df/dx = 6x−7.

  • Rule 5: The product rule

The derivative of the product y = u(x) × v(x), where u and v are both functions of x is

dy/dx = u × dv/dx + v × du/dx.

For example:

Differentiate f(x) = (6x2 + 2x) (x3 + 1).

Let u(x) = 6x2 + 2x and v(x) = x3 + 1.

Therefore, du/dx = 12x + 2 and dv/dx = 3x2.

Using the formula for the product rule, i.e. df /dx = u × dv/dx + v × du/dx

we get,

df/dx = (6x2 + 2x) (3x2) + (x3 + 1) (12x + 2),

          = 18x4 + 6x3 + 12x4 + 2x3 + 12x + 2,

          = 30x4 + 8x3 + 12x + 2.

  • Rule 6: The quotient rule

The derivative of the quotient f(x) = u(x) / v(x), where u and v are both function of x is

df/dx = (v × du/dx – u × dv/dx) / v2.

Example:

Differentiate f(x) = x2 + 7 / 3x − 1.

Let u(x) = x2 + 7 and v(x) = 3x − 1. Differentiate these to get du/dx = 2x and dv/dx = 3.

Now using the formula for the quotient rule we get,

df/dx = [(3x−1) (2x) − (x2 + 7) (3)] / (3x−1)2,

         = [6x2 − 2x − 3x2 – 21] / (3x−1)2,

     df/dx = [3x2 −2x−21] / (3x−1)2.

  • Rule 7: The chain rule

If y is a function of u, i.e. y = f(u), and u is a function of x, i.e. u = g(x) then the derivative of y with respect to x is

dy/dx =dy/du × du/dx.

Example:

Differentiate y = (x2 −5)4.

Let u = x2 − 5, therefore y = u4.

du/dx = 2x             and           ⇒ dy/du = 4u3.

Using the chain rule, we then get

dy/dx = dy/du × du/dx,

         = 4u3 ×2x,

        = 4(x2 −5)3 ×2x,

        = 8x (x2 −5)3.

 

In our next class, we will be talking about Integration and Evaluation of Simple Algebraic Functions. We hope you enjoyed the class.

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