Calculate Derivatives – Problems & Solutions

This screen contains only a summary of the rules needed to quickly calculate derivatives, and free problems for you to practice, each with a complete solution immediately available. If you need to quickly get this down, and don't have time to work through the more comprehensive material earlier in this Chapter, this screen is for you.

And if you're looking to review and practice for an exam, we suggest going to the next page on Chain Rule problems, since those are more advanced and encompass the use of all of the other rules.

Are you working to calculate derivatives in Calculus? Let's solve some common problems step-by-step so you can learn to solve them routinely for yourself.

Jump down this page to: [Power rule, 𝑥𝑛] [Exponential, 𝑒𝑥] [Trig derivatives] [Product rule] [Quotient rule] [Mixed problems] [Chain  rule]

CALCULUS SUMMARY: Derivatives and Rules

Let's step through each of the rules and give you the chance to try a few problems. Everything comes together, including more challenging problems, on the next screen regarding the Chain Rule.

I. Power Rule, 𝑥𝑛

Most frequently, you will use the Power Rule:

𝑑𝑑𝑥(𝑥𝑛)=𝑛𝑥𝑛−1
Power RulePractice Problem 1
Differentiate 𝑓(𝑥) =2𝜋.
Power Rule Practice Problem 2
Differentiate 𝑓(𝑥) =23𝑥9.
Power Rule Practice Problem 3
Differentiate 𝑓(𝑥) =2𝑥3 −4𝑥2 +𝑥 −33.
Power Rule Practice Problem 4
Differentiate 𝑓(𝑥) =𝑥1001 +5𝑥3 −6𝑥 +10,687.
Power Rule Practice Problem 5
Show that 𝑑𝑑𝑥√𝑥 =121√𝑥.
Recall that √𝑥 =𝑥1/2.
Power Rule Practice Problem 6
Differentiate 𝑓(𝑥) =5𝑥3.
Recall that 1𝑥𝑛 =𝑥−𝑛.
Power Rule Practice Problem 7
Differentiate 𝑓(𝑥) =1𝑥2 −4𝑥5.
Power Rule Practice Problem 8
Differentiate 𝑓(𝑥) =3√𝑥 −1√𝑥.
Power Rule Practice Problem 9
Differentiate 𝑓(𝑥) =√𝑥(𝑥2−8+1𝑥).
Power Rule Practice Problem 10
Differentiate 𝑓(𝑥) =(2𝑥2+1)2.

II. Exponential, 𝑒𝑥

𝑑𝑑𝑥𝑒𝑥=𝑒𝑥

This one's easy to remember!

Exponential Practice Problem 1
Differentiate 𝑓(𝑥) =𝑒𝑥 +𝑥.
Exponential Practice Problem 2
Differentiate 𝑓(𝑥) =𝑒1+𝑥.

More problems with exponentials are below in the Product Rule and Quotient Rule sections. And really, most problems involving the derivative of an exponential also require the Chain Rule, and those problems are on the next screen.

III. Trig Function Derivatives

𝑑𝑑𝑥(sin⁡𝑥)=cos⁡𝑥𝑑𝑑𝑥(csc⁡𝑥)=−csc⁡𝑥cot⁡𝑥𝑑𝑑𝑥(cos⁡𝑥)=−sin⁡𝑥𝑑𝑑𝑥(sec⁡𝑥)=sec⁡𝑥tan⁡𝑥𝑑𝑑𝑥(tan⁡𝑥)=sec2⁡𝑥𝑑𝑑𝑥(cot⁡𝑥)=−csc2⁡𝑥

Notice that a negative sign appears in the derivatives of the co-functions: cosine, cosecant, and cotangent.

Trig Function Practice Problem 1
Differentiate 𝑓(𝑥) =sin⁡𝑥 −cos⁡𝑥.
Trig Function Practice Problem 2
Differentiate 𝑓(𝑥) =5𝑥3 −tan⁡𝑥.

More problems with trig functions are below in the Product Rule and Quotient Rule sections. And then of course most problems involving the derivative of a trig function also requirethe Chain Rule so more problems are there.

IV. Product Rule

The Product Rule is used to find the derivative of the product of two functions:

𝑑𝑑𝑥(𝑓𝑔)=(𝑑𝑑𝑥𝑓)𝑔+𝑓(𝑑𝑑𝑥𝑔)=[ (derivative of the first) × (the second) ]+[ (the first) × (derivative of the second)]
Product Rule Practice Problem 1
Differentiate 𝑓(𝑥) =𝑥3𝑒𝑥.
Product Rule Practice Problem 2
Differentiate 𝑓(𝑥) =𝑥sin⁡𝑥.
(A) cos⁡𝑥(B) sin⁡𝑥−𝑥cos⁡𝑥(C) sin⁡𝑥+𝑥cos⁡𝑥
(D) −cos⁡𝑥(E) None of these
Product Rule Practice Problem 3
Differentiate 𝑔(𝜃) =sin⁡𝜃 cos⁡𝜃.
Product Rule Practice Problem 4
Differentiate 𝑓(𝑥) =(𝑒𝑥+1)tan⁡𝑥.
Product Rule Practice Problem 5
Differentiate 𝑧(𝑥) =𝑥5/2 𝑒𝑥sin⁡𝑥.
Product Rule Practice Problem 6
Given that 𝑓(2) =1, 𝑓′(2) = −3, 𝑔(2) =4, and 𝑔′(2) =8, find (𝑓𝑔)′(2).

V. Quotient Rule

The Quotient Rule is used to find the derivative of the quotient of two functions:

𝑑𝑑𝑥(𝑓𝑔)=(𝑑𝑑𝑥𝑓)𝑔−𝑓(𝑑𝑑𝑥𝑔)𝑔2=[(derivative of the numerator) × (the denominator)]−[ (the numerator) × (derivative of the denominator)]all divided by [the denominator, squared]

Many students remember the quotient rule by thinking of the numerator as "hi," the demoninator as "lo," the derivative as "d," and then singing

"lo d-hi minus hi d-lo over lo-lo"

Quotient Rule Practice Problem 1
Differentiate 𝑓(𝑥) =𝑥2𝑒𝑥.
Quotient Rule Practice Problem 2
Differentiate 𝑓(𝑥) =sin⁡𝑥𝑥.
Quotient Rule Practice Problem 3
Differentiate 𝑓(𝑥) =𝑒𝑥𝑥+1.
Quotient Rule Practice Problem 4
Differentiate 𝑓(𝑥) =3𝑥5−tan⁡𝑥.
Quotient Rule Practice Problem 5
Differentiate 𝑔(𝑢) =𝑢3−5𝑢2+6𝑢−2.
Stop after taking the derivatives; don't bother to multiply out the terms and simplify.
Quotient Rule Practice Problem 6
Given that 𝑓(2) =1, 𝑓′(2) = −3, 𝑔(2) =4, and 𝑔′(2) =8, find (𝑓𝑔)′(2).

VI. Mixed Problems

The problems below mix ideas from above, and include questions from actual university exams.

Mixed Problem 1

Find the requested information.

(a)
Let 𝑓(𝑢) =5𝑢−3𝑢2+1. Find 𝑓′(𝑢).
(b)
Let 𝑔(𝑡) =(𝑡2 +3 +𝑡−2)tan⁡𝑡. Find 𝑔′(𝑡).
Tip icon

Note: Problems similar to the following frequently appear on exams, and so we strongly urge you to learn how to solve it.

Mixed Problem 2
Find the values of 𝑎 and 𝑏 that will make the function 𝑓(𝑥) differentiable. 𝑓(𝑥)={𝑎𝑥+𝑏if 𝑥<𝜋sin⁡𝑥if 𝑥≥𝜋

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