Basic Exercises on the Derivative Function and Difference Quotients

1Problem 1

Use the definition to derive the derivative function of $$f\left(x\right)=x^3$$f(x)=x^3.

1.1Sample Solution

Let $$f:\mathbb{R}\to \mathbb{R}$$f:\mathbb{R}\to\mathbb{R} be defined by $$f\left(x\right)=x^3$$f(x)=x^3. For $$h\ne 0$$h\ne0,

Taking the limit as $$h\to 0$$h\to0 gives $$f^{\prime }\left(x\right)=3x^2$$f'(x)=3x^2.

1.2Explanation

Because the expression divides by $$h$$h, it must first be simplified under the condition $$h\ne 0$$h\ne0. Only then is the limit taken.

1.3Common Error

Substituting $$h=0$$h=0 before cancellation makes the difference quotient undefined.

2Problem 2

For $$f\left(x\right)=1/x$$f(x)=1/x, derive $$f^{\prime }\left(a\right)$$f'(a) from the definition, assuming $$a\ne 0$$a\ne0.

2.1Sample Solution

Let $$f:\mathbb{R}\setminus \left\{0\right\}\to \mathbb{R}$$f:\mathbb{R}\setminus\{0\}\to\mathbb{R}. For $$a\ne 0$$a\ne0, $$h\ne 0$$h\ne0, and $$a+h\ne 0$$a+h\ne0,

Therefore,

2.2Explanation

The denominators require all three conditions: $$a\ne 0$$a\ne0, $$h\ne 0$$h\ne0, and $$a+h\ne 0$$a+h\ne0.

2.3Common Error

A frequent error is to lose the negative sign when combining $$1/(a+h)-1/a$$1/(a+h)-1/a over a common denominator.

3Problem 3

Prove that $$f\left(x\right)=\left|x\right|$$f(x)=|x| is not differentiable at $$0$$0.

3.1Sample Solution

The difference quotient is

It equals $$1$$1 for $$h>0$$h>0 and $$-1$$-1 for $$h<0$$h<0. Hence,

Because the one-sided limits disagree, $$f^{\prime }\left(0\right)$$f'(0) does not exist, and $$f$$f is not differentiable at $$0$$0.

3.2Explanation

Continuity does not imply differentiability. A finite two-sided difference-quotient limit is required at an interior point.

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