Topics
>
Trigonometry>
Trigonometric Functions>
Derivative of Trigonometric Functions>
Derivative of Trigonometric Functions 3Derivative of Trigonometric Functions 3
Try VividMath Premium to unlock full access
Time limit: 0
Quiz summary
0 of 5 questions completed
Questions:
- 1
- 2
- 3
- 4
- 5
Information
–
You have already completed the quiz before. Hence you can not start it again.
Quiz is loading...
You must sign in or sign up to start the quiz.
You have to finish following quiz, to start this quiz:
Loading...
- 1
- 2
- 3
- 4
- 5
- Answered
- Review
-
Question 1 of 5
1. Question
Find the derivative using the product rule`tan3x cosx`Hint
Help VideoCorrect
Excellent!
Incorrect
Derivatives of Trigonometric Functions
`y'(\text(sin))=\text(cos)``y'(\text(cos))=-\text(sin)``y'(\text(tan))=\text(sec)^2`Product Rule
$$\frac{dy}{dx}=\color{#9a00c7}{v}\color{#e65021}{\frac{du}{dx}}+\color{#00880A}{u}\color{#004ec4}{\frac{dv}{dx}}$$First, find the derivative of `u` and `v`Derivative of `u`:`u` `=` `\text(tan) 3x` `u’` `=` `(du)/(dx)` `=` `3 sec^2 3x` Product Rule Derivative of `v`:`v` `=` `\text(cos) x` `v’` `=` `(dv)/(dx)` `=` `-\text(sin) x` Substitute the components into the product rule$$\frac{dy}{dx}$$ `=` $$\color{#9a00c7}{v}\color{#e65021}{\frac{du}{dx}}+\color{#00880A}{u}\color{#004ec4}{\frac{dv}{dx}}$$ `(dy)/(dx)` `=` $$(\color{#9a00c7}{\text{cos}\;x}\cdot\color{#e65021}{3\;sec^2\;3x})+[\color{#00880A}{\text{tan}\;3x}\cdot(\color{#004ec4}{-\text{sin}\;x})]$$ Substitute known values `y’` `=` `3\text(cos) 3x \text(sec)^2 3x-\text(sin) x \text(tan) 3x` Evaluate `y’=3\text(cos) 3x \text(sec)^2 3x-\text(sin) x \text(tan) 3x` -
Question 2 of 5
2. Question
Find the derivative using the product rule`8x sin(x/4)`Hint
Help VideoCorrect
Keep Going!
Incorrect
Derivatives of Trigonometric Functions
`y'(\text(sin))=\text(cos)``y'(\text(cos))=-\text(sin)``y'(\text(tan))=\text(sec)^2`Product Rule
$$\frac{dy}{dx}=\color{#9a00c7}{v}\color{#e65021}{\frac{du}{dx}}+\color{#00880A}{u}\color{#004ec4}{\frac{dv}{dx}}$$First, find the derivative of `u` and `v`Derivative of `u`:`u` `=` `8x` `u’` `=` `(du)/(dx)` `=` `8` Power Rule Derivative of `v`:`v` `=` `\text(sin) x/4` `v’` `=` `(dv)/(dx)` `=` `1/4\text(cos) x/4` Power Rule Substitute the components into the product rule$$\frac{dy}{dx}$$ `=` $$\color{#9a00c7}{v}\color{#e65021}{\frac{du}{dx}}+\color{#00880A}{u}\color{#004ec4}{\frac{dv}{dx}}$$ `(dy)/(dx)` `=` $$\left(\color{#9a00c7}{\text{sin}\;\frac{x}{4}}\cdot\color{#e65021}{8}\right)+\left(\color{#00880A}{8x}\cdot\color{#004ec4}{\frac{1}{4}}\text{cos}\;\frac{x}{4}\right)$$ Substitute known values `y’` `=` `8\text(sin) x/4+2x\text(cos) x/4` Evaluate `y’=8\text(sin) x/4+2x\text(cos) x/4` -
Question 3 of 5
3. Question
Find the derivative using the product rule`x^2 cos^3x`Hint
Help VideoCorrect
Great Work!
Incorrect
Derivatives of Trigonometric Functions
`y'(\text(sin))=\text(cos)``y'(\text(cos))=-\text(sin)``y'(\text(tan))=\text(sec)^2`Product Rule
$$\frac{dy}{dx}=\color{#9a00c7}{v}\color{#e65021}{\frac{du}{dx}}+\color{#00880A}{u}\color{#004ec4}{\frac{dv}{dx}}$$First, find the derivative of `u` and `v`Derivative of `u`:`u` `=` `x^2` `u’` `=` `(du)/(dx)` `=` `2x` Power Rule Derivative of `v`:`v` `=` `(\text(cos) x)^3` `v’` `=` `(dv)/(dx)` `=` `-3\text(cos)^2 x \text(sin) x` Chain Rule Substitute the components into the product rule$$\frac{dy}{dx}$$ `=` $$\color{#9a00c7}{v}\color{#e65021}{\frac{du}{dx}}+\color{#00880A}{u}\color{#004ec4}{\frac{dv}{dx}}$$ `(dy)/(dx)` `=` $$(\color{#9a00c7}{\text{cos}^3\;x}\cdot\color{#e65021}{2x})+(\color{#00880A}{x^2}\cdot\color{#004ec4}{-3\text{cos}^2x\;\text{sin}\;x})$$ Substitute known values `y’` `=` `2x\text(cos)^3 x-3x^2\text(cos)^2x \text(sin) x` Evaluate `y’=2x\text(cos)^3 x-3x^2\text(cos)^2x \text(sin) x` -
Question 4 of 5
4. Question
Find the derivative using the quotient rule`(tanx)/x`Hint
Help VideoCorrect
div class=”s-correct”>
Fantastic!
Incorrect
Derivatives of Trigonometric Functions
`y'(\text(sin))=\text(cos)``y'(\text(cos))=-\text(sin)``y'(\text(tan))=\text(sec)^2`Quotient Rule
$$\frac{dy}{dx}=\frac{\color{#9a00c7}{v}\color{#e65021}{\frac{du}{dx}}-\color{#00880A}{u}\color{#004ec4}{\frac{dv}{dx}}}{\color{#9a00c7}{v}^2}$$First, find the derivative of `u` and `v`Derivative of `u`:`u` `=` `\text(tan) x` `u’` `=` `(du)/(dx)` `=` `\text(sec)^2 x` Derivative of `v`:`v` `=` `x` `v’` `=` `(dv)/(dx)` `=` `1` Power Rule Substitute the components into the quotient rule$$\frac{dy}{dx}$$ `=` $$\frac{\color{#9a00c7}{v}\color{#e65021}{\frac{du}{dx}}-\color{#00880A}{u}\color{#004ec4}{\frac{dv}{dx}}}{\color{#9a00c7}{v}^2}$$ `(dy)/(dx)` `=` $$\frac{(\color{#9a00c7}{x}\cdot\color{#e65021}{\text{sec}^2\;x})-(\color{#00880A}{\text{tan}\;x}\cdot\color{#004ec4}{1})}{\color{#9a00c7}{x}^2}$$ Substitute known values `y’` `=` `(x \text(sec)^2 x-\text(tan) x)/(x^2)` Evaluate `y’=(x \text(sec)^2 x-\text(tan) x)/(x^2)` -
Question 5 of 5
5. Question
Find the derivative using the quotient rule`(sin3x)/(cos4x)`Hint
Help VideoCorrect
Exceptional!
Incorrect
Derivatives of Trigonometric Functions
`y'(\text(sin))=\text(cos)``y'(\text(cos))=-\text(sin)``y'(\text(tan))=\text(sec)^2`Quotient Rule
$$\frac{dy}{dx}=\frac{\color{#9a00c7}{v}\color{#e65021}{\frac{du}{dx}}-\color{#00880A}{u}\color{#004ec4}{\frac{dv}{dx}}}{\color{#9a00c7}{v}^2}$$First, find the derivative of `u` and `v`Derivative of `u`:`u` `=` `\text(sin) 3x` `u’` `=` `(du)/(dx)` `=` `3\text(cos) 3x` Chain Rule Derivative of `v`:`v` `=` `\text(cos) 4x` `v’` `=` `(dv)/(dx)` `=` `-4\text(sin) 4x` Chain Rule Substitute the components into the quotient rule$$\frac{dy}{dx}$$ `=` $$\frac{\color{#9a00c7}{v}\color{#e65021}{\frac{du}{dx}}-\color{#00880A}{u}\color{#004ec4}{\frac{dv}{dx}}}{\color{#9a00c7}{v}^2}$$ `(dy)/(dx)` `=` $$\frac{(\color{#9a00c7}{\text{cos}\;4x}\cdot\color{#e65021}{3\text{cos}\;3x})-[\color{#00880A}{\text{sin}\;3x}\cdot(\color{#004ec4}{-4\text{sin}\;4x})]}{\color{#9a00c7}{\text{cos}\;4x}^2}$$ Substitute known values `y’` `=` `(3\text(cos) 4x \text(cos)3x+4\text(sin) 3x \text(sin) 4x)/((\text(cos)4x)^2)` Evaluate `y’=(3\text(cos) 4x \text(cos)3x+4\text(sin) 3x \text(sin) 4x)/((\text(cos)4x)^2)`
Quizzes
- Derivative of Trigonometric Functions 1
- Derivative of Trigonometric Functions 2
- Derivative of Trigonometric Functions 3
- Trig Applications of Differentiation
- Integral of Trigonometric Functions 1
- Integral of Trigonometric Functions 2
- Trig Applications of Integration
- Graph Trigonometric Functions 1
- Graph Trigonometric Functions 2
- Graph Trigonometric Functions 3
- Graph Trigonometric Functions 4