Derivatives · Definition

Basic differentiation rules

calculusderivatives

Before you start you need to know

These rules let you differentiate polynomials and power functions without limits. They are the foundation for the rest of differential calculus.

Definition

The derivative of a constant is \(0\) and that of \(x^n\) is \(n\,x^{n-1}\). The derivative is also linear: the derivative of a sum is the sum of the derivatives and constants factor out.

\[\begin{gathered}\frac{d}{dx}x^{n}=nx^{n-1}\\ (cf+g)'=cf'+g'\end{gathered}\]

The power rule says \((x^{n})'=n\,x^{n-1}\) for any real exponent \(n\). A constant has derivative \(0\), and a constant multiple comes out of the derivative: \((c\,f)'=c\,f'\).

Example: for \(f(x)=3x^{4}-x^{2}+5\), we get \(f'(x)=12x^{3}-2x\). The \(3\) stays as a factor, \(x^{4}\) drops to \(x^{3}\) with a \(4\), and the \(5\) disappears.

Typical mistake: forgetting to subtract 1 from the exponent, writing \(x^{n}\) instead of \(x^{n-1}\), or treating the constant \(5\) as if it were \(5x\). Differentiate each term separately before adding.

Think about it

Differentiate \(3x^4-2x+7\).

\(12x^3-2\): derivative of each term; the constant 7 vanishes.

Exercises 10

Differentiate the following functions

  1. \(\displaystyle f(x)=5 x^{4} - 3 x^{2} + 7 x - 2\)
  2. \(\displaystyle f(x)=4 x^{3} + 2 x - 9\)
  3. \(\displaystyle f(x)=x^{5} - 3 x^{2} + 7\)
  4. \(\displaystyle f(x)=3 \sqrt{x} + \frac{1}{x^{2}}\)
  5. \(\displaystyle f(x)=x^{6}\)
  6. \(\displaystyle f(x)=\frac{7}{x^{3}}\)
  7. \(\displaystyle f(x)=-8\)
  8. \(\displaystyle f(x)=\frac{x^{3}}{2}+4 x\)

Problems

  1. Differentiate \(g(t)=\pi^{2}+6 t^{2}\).
  2. Differentiate \(h(x)=\sqrt[3]{x^{2}}+\frac{4}{x}\).

What comes next

References

  • James Stewart, Calculus: Early Transcendentals, 8.ª edición, Cengage Learning, 2015.