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Questions and Answers
What is the derivative of y = cos(f(x))?
What is the derivative of y = cos(f(x))?
- sin(f(x)) * f'(x)
- -cos(f(x)) * f'(x)
- -sin(f(x)) * f'(x) (correct)
- sec²(f(x)) * f'(x)
If y = tan(f(x)), which of the following represents its derivative?
If y = tan(f(x)), which of the following represents its derivative?
- sec(f(x)) * tan(f(x)) * f'(x)
- sec²(f(x)) * f'(x) (correct)
- -csc²(f(x)) * f'(x)
- sin²(f(x)) * f'(x)
What is the derivative of y = csc(f(x))?
What is the derivative of y = csc(f(x))?
- -sec(f(x)) * cot(f(x)) * f'(x)
- -csc(f(x))cot(f(x)) * f'(x) (correct)
- -csc(f(x)) * f'(x)
- sec(f(x))tan(f(x)) * f'(x)
Which derivative corresponds to y = sin(f(x))?
Which derivative corresponds to y = sin(f(x))?
What can be said about the derivative of y = sec(f(x))?
What can be said about the derivative of y = sec(f(x))?
What is the derivative of the function $F(x) = x^5$ using the power rule?
What is the derivative of the function $F(x) = x^5$ using the power rule?
Which expression represents the product rule for differentiating the function $f(x)g(x)$?
Which expression represents the product rule for differentiating the function $f(x)g(x)$?
What is the correct form of the quotient rule for the function $ rac{f(x)}{g(x)}$?
What is the correct form of the quotient rule for the function $ rac{f(x)}{g(x)}$?
Using the chain rule, what is the derivative of the function $(f(x))^3$?
Using the chain rule, what is the derivative of the function $(f(x))^3$?
What is the derivative of the function $F(x) = rac{e^{ ext{ln }x}}{x}$ at any point where $x > 0$?
What is the derivative of the function $F(x) = rac{e^{ ext{ln }x}}{x}$ at any point where $x > 0$?
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Study Notes
Derivatives of Trigonometric Functions
- The derivative of
sin(f(x))iscos(f(x)) * f'(x). - The derivative of
cos(f(x))is-sin(f(x)) * f'(x). - The derivative of
tan(f(x))issec²(f(x)) * f'(x). - The derivative of
cot(f(x))is-csc²(f(x)) * f'(x). - The derivative of
sec(f(x))issec(f(x))tan(f(x)) * f'(x). - The derivative of
csc(f(x))is-csc(f(x))cot(f(x)) * f'(x). - The
f'(x)notation represents the derivative off(x)with respect tox.
Differentiation Rules
- The power rule states:
F(x) = x^nimpliesd/dx F(x) = nx^(n-1). - The product rule states:
d/dx[f(x)g(x)] = f(x)g'(x) + g(x)f'(x). - The quotient rule states
d/dx[f(x)/g(x)] = [g(x)f'(x)-f(x)g'(x)]/[g(x)]^2. - The chain rule states
d/dx((f(x))^n) = n(f(x))^(n-1)f'(x). - The chain rule also states
d/dx√f(x) = f'(x)/(2√f(x)).
Example
- The derivative of
e^ln(x)ise^ln(x)/x.
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