Derivatives List & Flashcards

Derivatives List & Flashcards
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Differentiate y= 5x^2+ sin x cos x
Using the 'product rule' and the trigonometry identity, the answer is 10x+cos(2x).
Got it
Differentiate y= (1-4x+7x^5)^30

Using the 'chain rule,' differentiate the 30th power first, leaving (1-4x+7x^5) unchanged. Then differentiate (1-4x+7x^5).

Answer: 30(35x^4-4)(1-4x+7x^5)^29.

Got it
Differentiate y= sin(5x)

Using the 'chain rule,' differentiate the sine function first, leaving (5x) unchanged. Then differentiate (5x).

Answer: 5 cos (5x).

Got it
Differentiate y= (3x + 1)^2

Using the 'chain rule,' differentiate the square first, leaving (3x+1) unchanged. Then differentiate (3x+1).

Answer: 6 (3x+1)

Got it
Chain rule
The 'chain rule' computes the derivative of the composition of two or more functions. This rule is used when there are parentheses in a formula.
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Quotient rule
The 'quotient rule' describes when one function is divided by another. To remember, use the rhyme 'Low d hi minus hi d low, all over the square of what's below.'
Got it
Product rule
The 'product rule' describes when one function is multiplied by another function. When multiplying two powers with the same base, add the exponents.
Got it
Derivative
Either the slope of a plane or line or the rate of change of the function.
Got it
Tangent of x derivative
Secant squared of x (sec^2(x))
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Cos(x) derivative
-Sin(x)
Got it
Sin(x) derivative
Cos(x)
Got it
Natural log of x derivative
1/x
Got it
Exponential e^x derivative
e^x
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d/dx of any constant derivative
0
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(d/dx) of x derivative
1
Got it
x^nth power derivative
n times x^(n-1).
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32 cards in set

Flashcard Content Overview

Solving trigonometric functions can look like a challenge, but applying the rules of differentiation and using derivatives can help you solve just about any of them. Challenge your understanding of the rules and the common derivatives with these flashcards.

Front
Back
x^nth power derivative
n times x^(n-1).
(d/dx) of x derivative
1
d/dx of any constant derivative
0
Exponential e^x derivative
e^x
Natural log of x derivative
1/x
Sin(x) derivative
Cos(x)
Cos(x) derivative
-Sin(x)
Tangent of x derivative
Secant squared of x (sec^2(x))
Derivative
Either the slope of a plane or line or the rate of change of the function.
Product rule
The 'product rule' describes when one function is multiplied by another function. When multiplying two powers with the same base, add the exponents.
Quotient rule
The 'quotient rule' describes when one function is divided by another. To remember, use the rhyme 'Low d hi minus hi d low, all over the square of what's below.'
Chain rule
The 'chain rule' computes the derivative of the composition of two or more functions. This rule is used when there are parentheses in a formula.
Differentiate y= (3x + 1)^2

Using the 'chain rule,' differentiate the square first, leaving (3x+1) unchanged. Then differentiate (3x+1).

Answer: 6 (3x+1)

Differentiate y= sin(5x)

Using the 'chain rule,' differentiate the sine function first, leaving (5x) unchanged. Then differentiate (5x).

Answer: 5 cos (5x).

Differentiate y= (1-4x+7x^5)^30

Using the 'chain rule,' differentiate the 30th power first, leaving (1-4x+7x^5) unchanged. Then differentiate (1-4x+7x^5).

Answer: 30(35x^4-4)(1-4x+7x^5)^29.

Differentiate y= 5x^2+ sin x cos x
Using the 'product rule' and the trigonometry identity, the answer is 10x+cos(2x).
Differentiate y= (x^3+7x-1)(5x+2)
Using the 'product rule,' the answer is 20x^3+6x^2+70x+9.
Differentiate y=x^3 ln x
Using the 'product rule,' the answer is x^2(1+3lnx).
Differentiate y= 2 / x+1
Using the 'quotient rule,' the answer is -2 / (x+1)^2
Differentiate y= x^2 / 3x-1
Using the 'quotient rule,' the answer is x (3x-2) / (3x-1)^2.
Differentiate y= sin(x) / x^2
Using the 'quotient rule,' the answer is (xcos(x) - 2sin(x)) / x^3.
Differentiate y= 3z+9 / 2-z
Using the 'quotient rule,' the answer is 15 / (2-z)^2
Evaluate y= sin^-1(1/2)
y= Pi/6
Differentiate f(t)= 5(cos)^-1(t) - 10 tan^-1(t)
f^1(t)= -4/square root (1-t^2) - 10/ 1+t^2
Find f^1(x) if f(x)= cos^-1 (5-x)
f^1(x)= -5 / square root (1-25x^2)
Find y^1 if y=arctan(square root x^3)
y^1= 3 x square root x / 2(1+x^3)
Find the derivative of y= cos^-1 5x
y= -5/ square root (1-25x^2)
Find the derivative y= 4 tan ^-1 3x^4
y= 48x^3/ 1+9x^8
Solve f(x)=sin(x) / x^2
(xcos(x) - 2sin(x)) / x^3
Solve f(x)=e^x^2
f`(x)=2xe^(x^2)
Solve f(x)=(cos(x))^3
f`(x)= -3(sin(x)cos^2(x)
Solve f(x)=x^2 times the natural log (ln) of x
f`x=x + 2xln(x) or f`x=x(1 + 2ln(x))

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