# 3.E: Rules for Finding Derivatives (Exercises)

\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)

These are homework exercises to accompany David Guichard's "General Calculus" Textmap. Complementary General calculus exercises can be found for other Textmaps and can be accessed here.

## 3.1: The Power Rule

Find the derivatives of the given functions.

**Ex 3.1.1** \( x^{100}\) (answer)

**Ex 3.1.2** \( x^{-100}\) (answer)

**Ex 3.1.3** \(\displaystyle {1\over x^5}\) (answer)

**Ex 3.1.4** \( x^\pi\) (answer)

**Ex 3.1.5** \( x^{3/4}\) (answer)

**Ex 3.1.6** \( x^{-9/7}\) (answer)

3.2: Linearity of the Derivative

Find the derivatives of the functions in 1--6.

**Ex 3.2.1** \( 5x^3+12x^2-15\) (answer)

**Ex 3.2.2** \( -4x^5 + 3x^2 - 5/x^2\) (answer)

**Ex 3.2.3** \( 5(-3x^2 + 5x + 1)\) (answer)

**Ex 3.2.4** \(f(x)+g(x)\), where \( f(x)=x^2-3x+2\) and \( g(x)=2x^3-5x\) (answer)

**Ex 3.2.5** \( (x+1)(x^2+2x-3)\) (answer)

**Ex 3.2.6** \( \sqrt{625-x^2}+3x^3+12\) (See section __2.1__.) (answer)

**Ex 3.2.7 **Find an equation for the tangent line to \( f(x) = x^3/4 - 1/x\) at \(x=-2\). (answer)

**Ex 3.2.8** Find an equation for the tangent line to \( f(x)= 3x^2 - \pi ^3\) at \(x= 4\). (answer)

**Ex 3.2.9** Suppose the position of an object at time \(t\) is given by \( f(t)=-49 t^2/10+5t+10\). Find a function giving the speed of the object at time \(t\). The acceleration of an object is the rate at which its speed is changing, which means it is given by the derivative of the speed function. Find the acceleration of the object at time \(t\). (answer)

**Ex 3.2.10 **Let \( f(x) =x^3\) and \(c= 3\). Sketch the graphs of \(f\), \(cf\), \(f'\), and \((cf)'\) on the same diagram.

**Ex 3.2.11** The general polynomial \(P\) of degree \(n\) in the variable \(x\) has the form

\[ P(x)= \sum _{k=0 } ^n a_k x^k = a_0 + a_1 x + \ldots + a_n x^n\].

What is the derivative (with respect to \(x\)) of \(P\)? (answer)

**Ex 3.2.12** Find a cubic polynomial whose graph has horizontal tangents at \((-2 , 5)\) and \((2, 3)\). (answer)

**Ex 3.2.13** Prove that \( {d\over dx}(cf(x))= cf'(x)\) using the definition of the derivative.

**Ex 3.2.14 **Suppose that \(f\) and \(g\) are differentiable at \(x\). Show that \(f-g\) is differentiable at \(x\) using the two linearity properties from this section.

## 3.3: The Product Rule

In 1--4, find the derivatives of the functions using the product rule.

**Ex 3.3.1** \( x^3(x^3-5x+10)\) (answer)

**Ex 3.3.2** \( (x^2+5x-3)(x^5-6x^3+3x^2-7x+1)\) (answer)

**Ex 3.3.3** \( \sqrt{x}\sqrt{625-x^2}\) (answer)

**Ex 3.3.4** \(\displaystyle {\sqrt{625-x^2}\over x^{20}}\) (answer)

**Ex 3.3.5** Use the product rule to compute the derivative of \( f(x)=(2x-3)^2\). Sketch the function. Find an equation of the tangent line to the curve at \(x=2\). Sketch the tangent line at \(x=2\). (answer)

**Ex 3.3.6** Suppose that \(f\), \(g\), and \(h\) are differentiable functions. Show that \((fgh)'(x) = f'(x) g(x)h(x) + f(x)g'(x) h(x) + f(x) g(x) h'(x)\).

**Ex 3.3.7 **State and prove a rule to compute \((fghi)'(x)\), similar to the rule in the previous problem.

Remark 3.3.2 {Product notation} Suppose \( f_1 , f_2 , \ldots f_n\) are functions. The product of all these functions can be written \( \prod _{k=1 } ^n f_k.\) This is similar to the use of \( \sum\) to denote a sum. For example,

\[\prod _{k=1 } ^5 f_k =f_1 f_2 f_3 f_4 f_5\]

and

\[ \prod _ {k=1 } ^n k = 1\cdot 2 \cdot \ldots \cdot n = n!.\]

We sometimes use somewhat more complicated conditions; for example

\[\prod _{k=1 , k\neq j } ^n f_k\]

denotes the product of \( f_1\) through \( f_n\) except for \( f_j\). For example,

\[\prod _{k=1 , k\neq 4} ^5 x^k = x\cdot x^2 \cdot x^3 \cdot x^5 = x^{11}.\]

**Ex 3.3.8** The **generalized product rule** says that if \( f_1 , f_2 ,\ldots ,f_n\) are differentiable functions at \(x\) then

\[{d\over dx}\prod _{k=1 } ^n f_k(x) = \sum _{j=1 } ^n \left(f'_j (x) \prod _{k=1 , k\neq j} ^n f_k (x)\right).\]

Verify that this is the same as your answer to the previous problem when \(n=4\), and write out what this says when \(n=5\).

## 3.4: The Quotient Rule

Find the derivatives of the functions in 1--4 using the quotient rule.

**Ex 3.4.1** \( {x^3\over x^3-5x+10}\) (answer)

**Ex 3.4.2** \( {x^2+5x-3\over x^5-6x^3+3x^2-7x+1}\) (answer)

**Ex 3.4.3** \( {\sqrt{x}\over\sqrt{625-x^2}}\) (answer)

**Ex 3.4.4** \( {\sqrt{625-x^2}\over x^{20}}\) (answer)

**Ex 3.4.5 **Find an equation for the tangent line to \( f(x) = (x^2 - 4)/(5-x)\) at \(x= 3\). (answer)

**Ex 3.4.6** Find an equation for the tangent line to \( f(x) = (x-2)/(x^3 + 4x - 1)\) at \(x=1\). (answer)

**Ex 3.4.7** Let \(P\) be a polynomial of degree \(n\) and let \(Q\) be a polynomial of degree \(m\) (with \(Q\) not the zero polynomial). Using sigma notation we can write

\[P=\sum _{k=0 } ^n a_k x^k,\qquad Q=\sum_{k=0}^m b_k x^k. \]

Use sigma notation to write the derivative of the **rational function** \(P/Q\).

**Ex 3.4.8 **The curve \( y=1/(1+x^2)\) is an example of a class of curves each of which is called a **witch of Agnesi**. Sketch the curve and find the tangent line to the curve at \(x= 5\). (The word *witch* here is a mistranslation of the original Italian, as described at __http://mathworld.wolfram.com/WitchofAgnesi.html__ and __http://instructional1.calstatela.edu/sgray/Agnesi/WitchHistory/Historynamewitch.html__. (answer)

**Ex 3.4.9 **If \(f'(4) = 5\), \(g'(4) = 12\), \((fg)(4)= f(4)g(4)=2\), and \(g(4) = 6\), compute \(f(4)\) and \(\ds{d\over dx}{f\over g}\) at 4. (answer)

## 3.5: The Chain Rule

Find the derivatives of the functions. For extra practice, and to check your answers, do some of these in more than one way if possible.

**Ex 3.5.1** \( x^4-3x^3+(1/2)x^2+7x-\pi\) (answer)

**Ex 3.5.2** \( x^3-2x^2+4\sqrt{x}\) (answer)

**Ex 3.5.3** \( (x^2+1)^3\) (answer)

**Ex 3.5.4** \( x\sqrt{169-x^2}\) (answer)

**Ex 3.5.5** \( (x^2-4x+5)\sqrt{25-x^2}\) (answer)

**Ex 3.5.6** \( \sqrt{r^2-x^2}\), \(r\) is a constant (answer)

**Ex 3.5.7** \( \sqrt{1+x^4}\) (answer)

**Ex 3.5.8** \( {1\over\sqrt{5-\sqrt{x}}}\). (answer)

**Ex 3.5.9** \( (1+3x)^2\) (answer)

**Ex 3.5.10** \( {(x^2+x+1)\over(1-x)}\) (answer)

**Ex 3.5.11** \( {\sqrt{25-x^2}\over x}\) (answer)

**Ex 3.5.12** \( \sqrt{{169\over x}-x}\) (answer)

**Ex 3.5.13** \( \sqrt{x^3-x^2-(1/x)}\) (answer)

**Ex 3.5.14** \( 100/(100-x^2)^{3/2}\) (answer)

**Ex 3.5.15** \( {\root 3 \of{x+x^3}}\) (answer)

**Ex 3.5.16** \( \sqrt{(x^2+1)^2+\sqrt{1+(x^2+1)^2}}\) (answer)

**Ex 3.5.17** \( (x+8)^5\) (answer)

**Ex 3.5.18** \( (4-x)^3\) (answer)

**Ex 3.5.19** \( (x^2+5)^3\) (answer)

**Ex 3.5.20** \( (6-2x^2)^3\) (answer)

**Ex 3.5.21** \( (1-4x^3)^{-2}\) (answer)

**Ex 3.5.22** \( 5(x+1-1/x)\) (answer)

**Ex 3.5.23** \( 4(2x^2-x+3)^{-2}\) (answer)

**Ex 3.5.24** \( {1\over 1+1/x}\) (answer)

**Ex 3.5.25** \( {-3\over 4x^2-2x+1}\) (answer)

**Ex 3.5.26** \( (x^2+1)(5-2x)/2\) (answer)

**Ex 3.5.27** \( (3x^2+1)(2x-4)^3\) (answer)

**Ex 3.5.28** \( {x+1\over x-1}\) (answer)

**Ex 3.5.29** \( {x^2-1\over x^2+1}\) (answer)

**Ex 3.5.30** \( {(x-1)(x-2)\over x-3}\) (answer)

**Ex 3.5.31** \( {2x^{-1}-x^{-2}\over 3x^{-1}-4x^{-2}}\) (answer)

**Ex 3.5.32** \( 3(x^2+1)(2x^2-1)(2x+3)\) (answer)

**Ex 3.5.33** \( {1\over (2x+1)(x-3)}\) (answer)

**Ex 3.5.34** \( ((2x+1)^{-1}+3)^{-1}\) (answer)

**Ex 3.5.35** \( (2x+1)^3(x^2+1)^2\) (answer)

**Ex 3.5.36** Find an equation for the tangent line to \( f(x) = (x-2)^{1/3}/(x^3 + 4x - 1)^2\) at \(x=1\). (answer)

**Ex 3.5.37** Find an equation for the tangent line to \( y=9x^{-2}\) at \((3,1)\). (answer)

**Ex 3.5.38** Find an equation for the tangent line to \( (x^2-4x+5)\sqrt{25-x^2}\) at \((3,8)\). (answer)

**Ex 3.5.39** Find an equation for the tangent line to \( {(x^2+x+1)\over(1-x)}\) at \((2,-7)\). (answer)

**Ex 3.5.40** Find an equation for the tangent line to \( \sqrt{(x^2+1)^2+\sqrt{1+(x^2+1)^2}}\) at \( (1,\sqrt{4+\sqrt{5}})\). (answer)