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Wize University Calculus 1 Textbook > Derivatives

The Limit Definition of a Derivative

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The Limit Definition of the Derivative

The derivative is one of the most important and fundamental concepts in calculus. The process of differentiation tells us the rate of change of a function.


The Derivative (At a Point)

The derivative of a function f(x)f(x) at a point x=a,x=a, denoted f′(a)f'(a), is
f′(a)=lim⁡h→0f(a+h)−f(a)h\boxed{f'(a)=\displaystyle\lim_{h\rightarrow0}\frac{f(a+h)-f(a)}{h}}

When this limit exists, we say that the function is differentiable at x=a.x=a.

Note: The derivative f′(a)f'(a) is also denoted dfdx∣x=a\frac{df}{dx}\Bigg|_{x=a}


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Example: Definition of a Derivative


Using the definition of a derivative, find the derivative of f(x)=x.f(x)=\sqrt{x}.



f′(x)=lim⁡h→0x+h−xh⋅x+h+xx+h+x=lim⁡h→0x+h−xh(x+h+x)=lim⁡h→01x+h+x=12x\begin{array}{ll} f'(x) & =\displaystyle \lim_{h \to 0}\dfrac{\sqrt{x+h}-\sqrt{x}}{h}\cdot\dfrac{\sqrt{x+h}+\sqrt{x}}{\sqrt{x+h}+\sqrt{x}}\\ & =\displaystyle \lim_{h \to 0}\dfrac{x+h-x}{h(\sqrt{x+h}+\sqrt{x})}\\ & =\displaystyle \lim_{h \to 0}\dfrac{1}{\sqrt{x+h}+\sqrt{x}}\\ & =\displaystyle \bf \dfrac{1}{2\sqrt{x}} \end{array}



Using the definition of the derivative, find the derivative of f(x)=(x+2)2+xf(x)=(x+2)^2+x




When is Function Not Differentiable?

Here are the 4 cases where a function is not differentiable (i.e. the limit lim⁡h→0f(a+h)−f(a)h\displaystyle\lim_{h\rightarrow0}\frac{f(a+h)-f(a)}{h} does not exist)
  1. If a function is discontinuous at a point, then it is not differentiable at that point:
  2. Jump discontinuity: the function is not differentiable at the jump

  1. Hole (removable discontinuity): the function is not differentiable at the hole

  1. Vertical asymptote: the function is not differentiable at the x value where the vertical asymptote is
  1. A function is not differentiable at a corner or cusp: For example, the function f(x)=∣x∣f\left(x\right)=\left|x\right| is not differentiable at x=0x=0

  1. A function is not differentiable where there is a vertical tangent line: For example, the function f(x)=x13f\left(x\right)=x^{\frac{1}{3}} or x3\sqrt[3]{x} is not differentiable at x=0x=0

  1. A function is not differentiable if it oscillates back and forth as it approaches a certain point: For example, the function f(x)=sin⁡(1x)f(x)=\sin\left(\dfrac{1}{x}\right) is not differentiable at x=0x=0 because the ff value oscillates back and forth between 1 and -1.


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Checking Differentiability at a Point

A common differentiability question you might see on the exam is that they'll give you a piecewise function f(x)={g(x), x<ah(x), x≥af(x)=\begin{cases} g(x)&,~x<a\\ h(x)&,~x\ge a \end{cases} and ask if the function is differentiable at the connection point x=ax=a. For this function to be differentiable, we need to check that lim⁡h→0−f(a+h)−f(a)h=lim⁡h→0+f(a+h)−f(a)h\displaystyle \lim_{h\to 0^-}\dfrac{f(a+h)-f(a)}{h}=\displaystyle \lim_{h\to 0^+}\dfrac{f(a+h)-f(a)}{h}. However, this can be very time-consuming and hard to do. Instead, we can usually check the two following things:
  1. Check that lim⁡x→a−f(x)=lim⁡x→a+f(x)=f(a)\displaystyle \lim_{x\to a^-}f(x)=\lim_{x\to a^+}f(x)=f(a). Most of the time you can just check if g(a)=h(a)g(a)=h(a) (this is a commonly used trick that only works MOST of the time)
  2. Check that g′(a)=h′(a)g'(a)=h'(a). We can use differentiation rules to find g′(a)g'(a) and h′(a)h'(a) instead of having to evaluate a difficult limit. (See the next few sections to learn about these differentiation rules)
Extra Practice
Limit definition of the derivative (first principles)
If f(x)=1x,f(x)=\frac{1}{\sqrt{x}},then find the derivative of the function by first principles.
Definition of the Derivative
Use the definition of the derivative to compute the derivative of
f(x)=x+2x−1f(x) = x + \sqrt{2x - 1}
at x=1x = 1 .
Definition of the Derivative
Use the definition of the derivative to compute the derivative of
f(x)=x+2x−1f(x) = x + \sqrt{2x - 1}
at x=1x = 1 .
Limit Definition of a Derivative
Find the derivative of the function by first principles (i.e. using the limit definition of the derivative): h(t)=216−th\left(t\right)=2\sqrt{16-t}
Definition of the Derivative
Carefully state what the definition of the derivative of a function is at the point x=ax = a .
Definition of a Derivative
Using the definition of the derivative, find the derivative of f(x)=(x+2)2+xf(x)=(x+2)^2+x
Definition of a Derivative
Using the definition of the derivative, set up the derivative of f(x)=(x+2)2+xf(x)=(x+2)^2+x, but do not solve!
Limit Definition of a Derivative
Find a function f(x)f(x)and a number aa such that
f′(a)=lim⁡x→0cos⁡x−1xf^{\prime}(a)=\lim_{x\rightarrow 0}\frac{\cos{x}-1}{x}
Limit Definition of a Derivative
Given that f′(2)=4f^{\prime}(2)=4, find the following limit:
lim⁡x→2x−2f(x)−f(2) \lim_{x\rightarrow 2}\frac{\sqrt{x}-\sqrt{2}}{f(x)-f(2)}
Limit definition of a derivative
Find f′(x)f^{\prime}(x) using the limit definition of the derivative.
f(x)=1x+3 for x≠−3f(x)=\frac{1}{x+3}\,\text{for}\,x\neq -3
Definition of a Derivative
Using the definition of the derivative, find the derivative of f(x)=(x+2)2+xf(x)=(x+2)^2+x
Definition of the Derivative
Use the definition of the derivative to compute the derivative of
f(x)=x+2x−1f(x) = x + \sqrt{2x - 1}
at x=1x = 1 .
Limit Definition of a Derivative
Using the formal definition of a derivative, the derivative of f(x)=x2f\left(x\right)=\frac{\sqrt{x}}{2} simplifies to
Limit definition of the derivative (first principles)
If f(x)=1x,f(x)=\frac{1}{\sqrt{x}},then find the derivative of the function by first principles.
Practice: Definition of Derivative
Practice: Definition of Derivative
For which pair of f(x)f\left(x\right) and aa does the equality f(a)=lim⁡x→8 x+1−3x−8\displaystyle f\left(a\right)=\lim_{x\to8}\ \frac{\sqrt{x+1}-3}{x-8} hold?
Limit Definition of Derivatives
Which of the following limits represents a derivative?
i) lim⁡a→0 f(x+a)−f(x)x−a\displaystyle \lim_{a\to0}\ \frac{f\left(x+a\right)-f\left(x\right)}{x-a}
ii) lim⁡x→3 x−3x−3\displaystyle \lim_{x\to3}\ \frac{\sqrt{x-3}}{x-3}
Limit Definition of a Derivative
Using the formal definition of a derivative, the derivative of f(x)=x2f\left(x\right)=\frac{\sqrt{x}}{2} simplifies to
Limit definition of a derivative
Evaluate the following limit: lim⁡x→052030−(5+x)20302x\displaystyle\lim_{x\rightarrow0} \frac{5^{2030}-(5+x)^{2030}}{2x}
Limit Definition of a Derivative
Find the derivative of the function by first principles (i.e. using the limit definition of the derivative): h(t)=216−th\left(t\right)=2\sqrt{16-t}
Q.\textbf{Q.} Consider again the function f(x)=x2+1f(x)=x^2+1. What is the instantaneous rate of change of ff at x=2x=2? Answer using the definition of the derivative.
Q.\textbf{Q.} Show that f(x)=∣x−2∣−1f(x)=\lvert x-2\rvert-1 is not differentiable at x=2x=2.
Limit Definition of a Derivative: Working with Given Values
Given that f′(2)=4f^{\prime}(2)=4, find lim⁡x→2x−2f(x)−f(2)\displaystyle \lim_{x\rightarrow 2}\frac{\sqrt{x}-\sqrt{2}}{f(x)-f(2)}.
Limit Definition of a Derivative
Find a function ff and a number aa such that:
f′(a)=lim⁡x→0cos⁡x−1x{f^{\prime}(a)=\lim_{x\rightarrow 0}\frac{\cos{x}-1}{x}}
Limits: Definition with Fractions
Find the derivative of g(x)=12+x\displaystyle g(x)=\frac{1}{2+x} using the definition.
Limit Definition of a Derivative
Q:\textbf{Q:} Find f′(2)f'(2) if f(x)=(2x+5)12f(x)=(2x+5)^{\frac{1}{2}} using the definition of the derivative.
Limit Definition of a Derivative
(Original question is written in long answer. Multiple choice has been used here for convenience)
Let f(x)=xx−3\displaystyle f(x) = \frac{x}{x - 3}. Compute dfdx\displaystyle \frac{df}{dx} using the definition of the derivative. No marks will be given for the use of derivative rules, but you may use them to check your answer.
Infinite Limits and Continuity
Given the function f(x)={arctan⁡x+π/2if x≤02xif 0<x≤e1ln⁡xif x>ef(x)=\begin{cases} \arctan x+\pi/2&\text{if }x \le 0\\ \frac{2}{x}&\text{if }0<x\le e\\ \frac{1}{\ln x}&\text{if }x>e \end{cases}, which of the following statements is true about f(x)f(x)?
Limit Definition of a Derivative
Find a function f(x)f(x)and a number aa such that
f′(a)=lim⁡x→0cos⁡x−1xf^{\prime}(a)=\lim_{x\rightarrow 0}\frac{\cos{x}-1}{x}
Limit Definition of a Derivative
Find the following derivatives using the definition of derivative.
f(x)=2x3+1f(x)=2x^3+1
Find f′(x)f^{\prime}(x) using the limit definition of the derivative if f(x)=1x2+3\displaystyle f(x)=\frac{1}{x^2+3}
Find the following using the definition of derivative.
f(x)=x3f(x)=\sqrt[3]{x}
Find f′(x)f^{\prime}(x) using the limit definition of the derivative.
f(x)=1x+3 for x≠−3f(x)=\frac{1}{x+3}\,\text{for}\,x\neq -3
Limit Definition of a Derivative
Evaluate
ddx(−2x2−2x+3)\dfrac{d}{dx}(-2x^2-2x+3)
using the limit definition of derivative.
Limit Definition of a Derivative
Evaluate ddx(−x2+7x−2)\dfrac{d}{dx}(-x^2+7x-2) using the limit definition of derivative.
Limit Definition of a Derivative
Given that f′(2)=4f'(2)=4, find
lim⁡x→2x−2f(x)−f(2)\lim\limits_{x\to 2} \frac{\sqrt{x}-\sqrt{2}}{f(x)-f(2)}
Limit Definition of a Derivative
If f(x) is differentiable everywhere, find A and B.
f(x)={x2+1x≥0Ax+Bx<0f(x)=\begin{cases}x^2+1 & x \geq 0\\ Ax+B&x<0\end{cases}
Limit Definition of a Derivative
Find a function f and a number a such that
f′(a)=lim⁡x→0cos⁡x−1xf'(a)=\lim\limits_{x\to 0}\frac{\cos x-1}{x}
Derivatives: Limit Definition
True or false: f(x)=∣x−3∣−1f(x) = |x-3|-1 is differentiable at x=3.x=3.
Limit definition of a derivative
Find f′(2)f'(2) if 2f(2)−2f(2+h)h=3h−2\frac{2f(2)-2f(2+h)}{h}=\frac{3}{h-2} for all h near 0.
Practice: Differentiability
Practice: Differentiability
Which of the following is/are NOT differentiable at x=0x=0?
(Select all that apply)
Find the derivative of f(x)=1x+3\displaystyle f(x)=\frac{1}{x+3}, using the limit definition.
Limit Definition of a Derivative
Use the limit definition to find the derivative of g(x)=xg(x)=\sqrt{x}.
Limit Definition of a Derivative
Given that f′(2)=4f^{\prime}(2)=4, find the following limit:
lim⁡x→2x−2f(x)−f(2) \lim_{x\rightarrow 2}\frac{\sqrt{x}-\sqrt{2}}{f(x)-f(2)}
Limit Definition of a Derivative
Find f′(x)f^{\prime}(x) using the limit definition of the derivative if f(x)=1x2+3\displaystyle f(x)=\frac{1}{x^2+3}
Limit definition of a derivative
Evaluate the following limit: lim⁡x→052030−(5+x)20302x\displaystyle\lim_{x\rightarrow0} \frac{5^{2030}-(5+x)^{2030}}{2x}
Limit Definition of a Derivative
Find the derivative of the function by first principles (i.e. using the limit definition of the derivative): h(t)=216−th\left(t\right)=2\sqrt{16-t}
Q.\textbf{Q.} Consider again the function f(x)=x2+1f(x)=x^2+1. What is the instantaneous rate of change of ff at x=2x=2? Answer using the definition of the derivative.
Q.\textbf{Q.} Show that f(x)=∣x−2∣−1f(x)=\lvert x-2\rvert-1 is not differentiable at x=2x=2.
Limit Definition of a Derivative: Working with Given Values
Given that f′(2)=4f^{\prime}(2)=4, find lim⁡x→2x−2f(x)−f(2)\displaystyle \lim_{x\rightarrow 2}\frac{\sqrt{x}-\sqrt{2}}{f(x)-f(2)}.
Limit Definition of a Derivative
Find a function ff and a number aa such that:
f′(a)=lim⁡x→0cos⁡x−1x{f^{\prime}(a)=\lim_{x\rightarrow 0}\frac{\cos{x}-1}{x}}
Limit Definition of a Derivative
Q:\textbf{Q:} Find f′(2)f'(2) if f(x)=(2x+5)12f(x)=(2x+5)^{\frac{1}{2}} using the definition of the derivative.
Limits: Definition with Fractions
Find the derivative of g(x)=12+x\displaystyle g(x)=\frac{1}{2+x} using the definition.
🦊 TRICKY! If f(x)f(x) is differentiable at x=1x=1 and f′(1)=4f'(1)=4 then find the limit lim⁡x→14x3−4x2f(x)−f(1)\displaystyle \lim_{x\rightarrow1}\frac{4x^3-4x^2}{f(x)-f(1)}.
Limit Definition of a Derivative
The given limit represents the derivative of the function f(x)f\left(x\right) at a value aa. Identify the function and the value of aa.
lim⁡h →0 sin⁡(π2+h)−1h\displaystyle\lim_{h\ \to0\ }\frac{\sin\left(\frac{\pi}{2}+h\right)-1}{h}
Limit Definition of a Derivative
Find the derivative of f(x)=13x+1\displaystyle f(x) = \frac{1}{3x + 1} by the definition of the derivative.
Calculate the derivative of the function by first principles (i.e. using the limit definition of the derivative)
h(t)=216−th\left(t\right)=2\sqrt{16-t}
Find f′(2) if f(x)=(2x+5)12f'(2)\ \text{if}\ f(x)=(2x+5)^{\frac{1}{2}} using the definition of the derivative.
Using the definition of the derivative, find f′(x)f'\left(x\right) for f(x)=x2−1f(x)=x^2−1.
Limit definition of the derivative (first principles)
If f(x)=1x,f(x)=\frac{1}{\sqrt{x}},then find the derivative of the function by first principles.
Limit Definition of a Derivative
Find f′(x)f^{\prime}(x) using the limit definition of the derivative if f(x)=2x+3\displaystyle f(x)=\sqrt{2x+3}
Limit Definition of a Derivative
Evaluate the following limit by recognizing it as a derivative of a function at a certain point.
lim⁡x→1 ln⁡x1−x\lim_{x\to1}\ \frac{\ln x}{1-x}.
Practice: Definition of Derivative
Practice: Definition of Derivative
For which pair of f(x)f\left(x\right) and aa does the equality f(a)=lim⁡x→8 x+1−3x−8\displaystyle f\left(a\right)=\lim_{x\to8}\ \frac{\sqrt{x+1}-3}{x-8} hold?
Limit Definition of Derivatives
Which of the following limits represents a derivative?
i) lim⁡a→0 f(x+a)−f(x)x−a\displaystyle \lim_{a\to0}\ \frac{f\left(x+a\right)-f\left(x\right)}{x-a}
ii) lim⁡x→3 x−3x−3\displaystyle \lim_{x\to3}\ \frac{\sqrt{x-3}}{x-3}
Limit Definition of a Derivative
Using the formal definition of a derivative, the derivative of f(x)=x2f\left(x\right)=\frac{\sqrt{x}}{2} simplifies to
Let f(x)=14−5xf(x) = \frac{1}{4 - 5x} . Use the definition of the derivative to compute f′(3)f'(3) . No credit will be given for the use of any differentiation rule.
Let f(x)=14−5xf(x) = \frac{1}{4 - 5x} . Use the definition of the derivative to compute f′(3)f'(3) . No credit will be given for the use of any differentiation rule.
Limit definition of a derivative
Using the definition of the derivative, solve for the derivative of f(x)f(x) at the point x=2x = 2 :
f(x)=13x+1−1f(x)= \frac{1}{\sqrt{3x + 1} - 1}
Limit Definition of a Derivative
State the definition of the derivative of a function f(x)f(x) at a point x=ax = a .
Definition of the Derivative
Use the definition of the derivative to compute the derivative of
f(x)=x+2x−1f(x) = x + \sqrt{2x - 1}
at x=1x = 1 .
Definition of the Derivative
Carefully state what the definition of the derivative of a function is at the point x=ax = a .
Definition of a Derivative
Using the definition of the derivative, find the derivative of f(x)=(x+2)2+xf(x)=(x+2)^2+x
Limit definition of a derivative
Find f′(x)f^{\prime}(x) using the limit definition of the derivative.
f(x)=1x+3 for x≠−3f(x)=\frac{1}{x+3}\,\text{for}\,x\neq -3
Find f′(x)f^{\prime}(x) using the limit definition of the derivative if f(x)=1x2+3\displaystyle f(x)=\frac{1}{x^2+3}