Derivatives · Definition

Derivative

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Before you start you need to know

The derivative is the central tool of calculus for measuring how a function changes at a point.

Definition

The derivative of \(f\) at \(a\) is the limit of the difference quotient as \(h\to0\). It measures the instantaneous rate of change and is the slope of the tangent line at \((a,f(a))\).

\[f'(a)=\lim_{h\to0}\frac{f(a+h)-f(a)}{h}\]

The derivative measures how fast a function changes at a point. It is defined as the limit of the difference quotient: the slope of the secant line as the two points merge, \(f'(a)=\lim_{h\to0}\frac{f(a+h)-f(a)}{h}\).

It has many readings depending on context: slope of the tangent (geometry), instantaneous velocity (physics), rate of change (economics, biology). It is the same idea under different names.

A function can be continuous at a point without being differentiable (for example, the corner of |x| at 0). Differentiable implies continuous, but not the other way around.

Think about it

Is \(|x|\) differentiable at 0?

No: the left-hand limit of the difference quotient is −1 and the right-hand one is 1; they differ.

Is \(f(x)=x^2\sin(1/x)\), with \(f(0)=0\), differentiable at \(0\)?

Yes, with \(f'(0)=0\): the quotient is \(h\sin(1/h)\), which is bounded by \(|h|\to0\).

Exercises 20

  1. The number \(f'(a)\) represents...
  2. If a function is differentiable at a, then...
  3. Using the limit definition of the derivative, what is \(f'(2)\) if \(f(x)=x^{2} + x\)?
  4. Using the limit definition of the derivative, what is \(f'(1)\) if \(f(x)=3 x^{2} - 2 x\)?
  5. Using the limit definition of the derivative, what is \(f'(2)\) if \(f(x)=x^{3}\)?
  6. Using the limit definition of the derivative, what is \(f'(4)\) if \(f(x)=\sqrt{x}\)?
  7. Using the limit definition of the derivative, what is \(f'(2)\) if \(f(x)=\frac{1}{x}\)?
  8. Which limit defines the derivative of \(f\) at \(a\)?
  9. Let \(f(x)=|x|\). What happens with \(f'(0)\)?
  10. If \(f(x)=c\) with \(c\) a constant, what is \(f'(x)\)?
  11. Compute the derivative of \(f(x)=x^2\) at \(c=3\) from the definition.
  12. Compute \(f'(2)\) for \(f(x)=1/x\).
  13. Compute \(f'(4)\) for \(f(x)=\sqrt x\).
  14. Interpret \(\displaystyle\lim_{x\to2}\frac{x^3-8}{x-2}\) as a derivative and compute it.
  15. For \(f(x)=x^3\) at \(c=1\), what does the difference quotient \(\dfrac{f(1+h)-f(1)}{h}\) equal for \(h\ne0\)?
  16. For \(f(x)=|x|\), what happens with \(f'(0)\)?
  17. For \(f(x)=x|x|\), what is \(f'(0)\)?
  18. The tangent line to \(f(x)=x^2\) at the point with abscissa \(1\) is:
  19. Let \(f(x)=x^{1/3}\) (real cube root). What happens with \(f'(0)\)?
  20. Let \(f(x)=x^2\) if \(x\le1\) and \(f(x)=ax+b\) if \(x>1\). For which \(a,b\) does \(f'(1)\) exist?

References

  • James Stewart, Calculus: Early Transcendentals, 8.ª edición, Cengage Learning, 2015.
  • Michael Spivak, Calculus, 4.ª edición, Publish or Perish, 2008.
  • Stephen Abbott, Understanding Analysis, 2nd edition, Springer, 2015. §5.2, Definición 5.2.1
  • Walter Rudin, Principles of Mathematical Analysis, 3rd edition, McGraw-Hill, 1976. Cap. 5, Definición 5.1
  • Robert G. Bartle, Donald R. Sherbert, Introduction to Real Analysis, 4th edition, Wiley, 2011. §6.1, Definición 6.1.1