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Question 1 of 5
Find the derivative using the chain rule
(3x+tan 7x)6
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Derivatives of Trigonometric Functions
Chain Rule
y′=n⋅(f(x))n−1⋅f′(x)
First, identify the values of the function
f(x) |
= |
xn |
f(x) |
= |
(3x+tan7x)6 |
Finally, substitute the values into the chain rule
y’ |
= |
n⋅(f(x))n−1⋅f′(x) |
|
|
= |
6⋅(3x+tan7x)6−1⋅f′(3x+tan7x) |
Substitute known values |
|
|
= |
6(3x+tan7x)5⋅3+sec27x⋅7 |
Differentiate the values |
|
|
= |
6(3x+tan7x)5(3+7sec27x) |
Evaluate |
y’=6(3x+tan 7x)5(3+7sec2 7x)
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Question 2 of 5
Find the derivative using the chain rule
sin-1x
Incorrect
Derivatives of Trigonometric Functions
Chain Rule
y′=n⋅(f(x))n−1⋅f′(x)
The expression can also be written as
Next, identify the values of the function
Finally, substitute the values into the chain rule
y’ |
= |
n⋅(f(x))n−1⋅f′(x) |
|
|
= |
−1⋅(sinx)−1−1⋅f′(sinx) |
Substitute known values |
|
|
= |
(−sinx)−2⋅cosx |
Differentiate the values |
|
|
= |
−cosxsin2x |
Reciprocate sin2x |
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Question 3 of 5
Find the derivative using the chain rule
(cos x+sin x)2
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Derivatives of Trigonometric Functions
Chain Rule
y′=n⋅(f(x))n−1⋅f′(x)
First, identify the values of the function
f(x) |
= |
xn |
f(x) |
= |
(cosx+sinx)2 |
Finally, substitute the values into the chain rule
y’ |
= |
n⋅(f(x))n−1⋅f′(x) |
|
|
= |
2⋅(cosx+sinx)2−1⋅f′(cosx+sinx) |
Substitute known values |
|
|
= |
2(cosx+sinx)(−sinx+cosx) |
Differentiate the values |
|
|
= |
2(cos2x−sin2x) |
Evaluate |
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Question 4 of 5
Find the derivative using the chain rule
3sin2 3x
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Derivatives of Trigonometric Functions
Chain Rule
y′=n⋅(f(x))n−1⋅f′(x)
The expression can also be written as
Remove the denominator by reciprocating its value
3(sin 3x)2 |
= |
3(sin 3x)-2 |
Reciprocate the denominator |
Next, identify the values of the function
f(x) |
= |
xn |
f(x) |
= |
3(sin3x)−2 |
Finally, substitute the values into the chain rule
y’ |
= |
n⋅(f(x))n−1⋅f′(x) |
|
|
= |
3⋅−2⋅(sin3x)−2−1⋅f′(sin3x) |
Substitute known values |
|
|
= |
−6(sin3x)−3⋅cos3x⋅3 |
Differentiate the values |
|
|
= |
−6⋅3cos3xsin33x |
Reciprocate (sin 3x)-3 |
|
|
= |
−18cos3xsin33x |
Evaluate |
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Question 5 of 5
Find the derivative using the product rule
x4cos3x
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Derivatives of Trigonometric Functions
Product Rule
dydx=vdudx+udvdx
First, find the derivative of u and v
Derivative of u:
u |
= |
x4 |
u’ |
= |
dudx |
= |
4x3 |
Power Rule |
Derivative of v:
v |
= |
cos 3x |
v’ |
= |
dvdx |
= |
-sin 3x⋅3 |
Chain Rule |
Substitute the components into the product rule
dydx |
= |
vdudx+udvdx |
|
dydx |
= |
(cos3x⋅4x3)+(x4⋅−sin3x⋅3) |
Substitute known values |
y’ |
= |
4x3cos 3x-3x4sin 3x |
Evaluate |