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Quizzes > Mathematics > Advanced Math

Ready to Master Trig Derivatives & Antiderivatives?

Think you can conquer antiderivatives and derivatives of trig functions? Start the quiz!

Difficulty: Moderate
2-5mins
Learning OutcomesCheat Sheet
Paper art quiz on teal background with free trig derivative and antiderivative problems csc x cot x sin x cos x challenge

This quiz helps you practice trig derivatives and antiderivatives with clear steps and quick checks. Work through problems like the antiderivative of csc x cot x and the derivative of sin(x)cos(x), and use instant feedback to spot gaps before a test and lock in key trig rules.

What is the derivative of sin(x)?
-cos(x)
sin(x)
cos(x)
-sin(x)
The derivative of sin(x) with respect to x is cos(x), derived from the limit definition of derivative and fundamental trigonometric limits. This relationship is one of the core results in differential calculus for trig functions. Knowing this allows you to solve many related rates and motion problems. For more detail, see .
What is the derivative of cos(x)?
-cos(x)
sin(x)
-sin(x)
cos(x)
By the standard differentiation rules, the derivative of cos(x) is -sin(x). This comes directly from the limit approach and mirror symmetry properties of sine and cosine. Understanding this helps in solving integrals and derivatives involving phase shifts. See more at .
What is the antiderivative of cos(x)?
-cos(x) + C
-sin(x) + C
sin(x) + C
cos(x) + C
Integrating cos(x) with respect to x gives sin(x) plus the constant of integration C, since differentiation of sin(x) returns cos(x). This is a fundamental antiderivative in calculus courses. It is widely used to revert back from rates of change to original functions. More at .
What is the antiderivative of sec²(x)?
-tan(x) + C
cot(x) + C
tan(x) + C
sec(x) + C
Since the derivative of tan(x) is sec²(x), its antiderivative is tan(x) plus the integration constant C. This result follows from the chain rule in reverse. Recognizing these standard pairs speeds up solving many integral problems. More details can be found at .
What is the derivative of tan(x)?
sec(x)
sec²(x)
sec(x)tan(x)
tan²(x)
The derivative of tan(x) is sec²(x), a result obtained by applying the quotient rule to sin(x)/cos(x). This identity is critical in solving differential equations and in physics for motion analysis. A concise proof and more examples are available at .
What is the derivative of sin(x)·cos(x)?
-cos(2x)
sin(2x)
cos(2x)
-sin(2x)
Using the product rule, d[sin(x)cos(x)] = cos²(x) - sin²(x), which simplifies via a double-angle identity to cos(2x). Recognizing this simplifies many trigonometric derivative problems. For more on product rule applications, see .
What is the antiderivative of csc(x)·cot(x)?
csc(x) + C
-ln|csc(x)+cot(x)| + C
-csc(x) + C
ln|csc(x)-cot(x)| + C
Since the derivative of csc(x) is -csc(x)cot(x), integrating csc(x)cot(x) yields -csc(x) + C. Recognizing these derivative pairs is key for quick antiderivative computation. For a full list of integrals, visit .
What is the derivative of sec(x)?
sec(x)
sec(x)·tan(x)
tan²(x)
sec²(x)
Differentiating sec(x) = 1/cos(x) via the quotient rule or chain rule gives sec(x)tan(x). This identity is essential in advanced calculus and physics. See for details.
What is the antiderivative of 1/(1+x²)?
arctan(x) + C
x/(1+x²) + C
arccot(x) + C
ln(1+x²) + C
The integral of 1/(1+x²) is arctan(x) + C, because the derivative of arctan(x) yields that integrand. This is a standard result in calculus courses. More on this transform is at .
What is the derivative of arctan(x)?
1/(x²)
1/(1+x²)
x/(1+x²)
1/(1-x²)
By implicit differentiation of y = arctan(x), one finds dy/dx = 1/(1+x²). This formula is heavily used in inverse-trig integration and differential equations. For more, see .
What is the antiderivative of sin²(x)?
-x/2 + sin(2x)/4 + C
x/2 + sin(2x)/4 + C
x/2 - sin(2x)/4 + C
x/2 - cos(2x)/4 + C
Using the identity sin²(x) = (1 - cos(2x))/2, integrate to get x/2 - (sin(2x))/4 + C. This reduction via double-angle identities is common in advanced integral problems. More examples at .
What is the derivative of cos(3x)?
-sin(3x)
3·cos(3x)
3·sin(3x)
-3·sin(3x)
Applying the chain rule, d[cos(3x)] = -sin(3x)·3 = -3 sin(3x). This factor of 3 arises because the inner function 3x differentiates to 3. For more on chain rule applications, see .
What is the antiderivative of tan(x)?
ln|cos(x)| + C
ln|sec(x)| + C
-ln|cos(x)| + C
-ln|sec(x)| + C
Rewriting tan(x) as sin(x)/cos(x) and using substitution u = cos(x) leads to -ln|cos(x)| + C. This is a classic result in integral calculus. An alternative form is ln|sec(x)| + C, though only one correct format is accepted here. See .
What is the derivative of arccos(x)?
1/(1 - x²)
-1/?(1 - x²)
-1/(1 + x²)
1/?(1 - x²)
Differentiating y = arccos(x) yields dy/dx = -1/?(1 - x²) by implicit differentiation and Pythagorean identity. This negative sign distinguishes arccos from arcsin. More at .
What is the antiderivative of sec(x)?
sec(x)·tan(x) + C
ln|sec(x) + tan(x)| + C
tan(x) + C
ln|sec(x) - tan(x)| + C
A standard trick multiplies sec(x) by (sec(x)+tan(x))/(sec(x)+tan(x)) and integrates via substitution to get ln|sec(x)+tan(x)| + C. This integral appears in many engineering and physics contexts. For a derivation, see .
What is the antiderivative of sec³(x)?
½[sec(x)·tan(x) - ln|sec(x)+tan(x)|] + C
½[sec(x)·tan(x) + tan(x)] + C
?[sec(x)·tan(x) + ln|sec(x)+tan(x)|] + C
½[sec(x)·tan(x) + ln|sec(x)+tan(x)|] + C
Integrating sec³(x) uses integration by parts with u = sec(x) and dv = sec²(x)dx, yielding ½[sec(x)tan(x)+ln|sec(x)+tan(x)|] + C. This is one of the more challenging standard trig integrals. For full steps, see .
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Study Outcomes

  1. Understand Trig Derivative Rules -

    Understand fundamental differentiation rules for sine, cosine, and other trig functions to confidently tackle trig derivatives and antiderivatives problems.

  2. Apply Antiderivative Techniques -

    Apply integration strategies to determine the cscxcotx antiderivative and reinforce mastery of antiderivatives and derivatives of trig functions.

  3. Analyze Product Derivatives -

    Analyze products such as sin(x) cos(x) to derive formulas and practice calculating the derivative of sin(x) cos(x) with precision.

  4. Solve Quiz Problems Efficiently -

    Solve a variety of quiz questions covering both trig derivatives and antiderivatives with step-by-step reasoning for quick and accurate results.

  5. Evaluate Proficiency -

    Evaluate your understanding through immediate feedback, identifying areas for improvement in your trig calculus skills.

  6. Master Advanced Techniques -

    Master advanced substitution and integration methods to expand your toolkit for tackling more complex trigonometric integrals.

Cheat Sheet

  1. Basic Derivative Rules -

    The derivatives of sin(x) and cos(x) form the foundation of trig derivatives and antiderivatives; according to Stewart Calculus and MIT OpenCourseWare, d/dx[sin(x)] = cos(x) and d/dx[cos(x)] = −sin(x). A handy mnemonic is "Sine's co-derivative is cosine," helping you lock in these essential formulas.

  2. Derivatives of tan, cot, sec & csc -

    When exploring the derivatives of tan, cot, sec, and csc, you'll see how these fit into the broader antiderivatives and derivatives of trig functions framework: d/dx[tan(x)]=sec²(x), d/dx[cot(x)]=−csc²(x), d/dx[sec(x)]=sec(x)·tan(x), and d/dx[csc(x)]=−csc(x)·cot(x). A common mnemonic is "All Students Take Calculus" to remember the order of positive functions in each quadrant.

  3. Product Rule & sin(x)·cos(x) -

    To find the derivative of sin(x)·cos(x), apply the product rule or use the double-angle identity: sin(x)cos(x)=½sin(2x). Differentiating gives d/dx[sin(x)cos(x)] = cos²(x) − sin²(x) = cos(2x), a trick often featured in university coursework.

  4. Antiderivatives of sin & cos -

    Working backwards from derivatives, the antiderivatives are ∫sin(x)dx = −cos(x)+C and ∫cos(x)dx = sin(x)+C, fundamental to many physics and engineering problems. Remembering that integration is "differentiation in reverse" helps you build confidence when tackling more complex integrals of trig expressions.

  5. csc(x)·cot(x) Antiderivative -

    The integral ∫csc(x)·cot(x)dx = −csc(x)+C follows directly from knowing d/dx[csc(x)] = −csc(x)·cot(x), a key result in both research and standard calculus texts. Keeping these paired derivative-antiderivative relationships in mind streamlines your workflow when reviewing trig integrals.

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