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Questions and Answers
What is the derivative of y = cos(f(x))?
If y = tan(f(x)), which of the following represents its derivative?
What is the derivative of y = csc(f(x))?
Which derivative corresponds to y = sin(f(x))?
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What can be said about the derivative of y = sec(f(x))?
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What is the derivative of the function $F(x) = x^5$ using the power rule?
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Which expression represents the product rule for differentiating the function $f(x)g(x)$?
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What is the correct form of the quotient rule for the function $ rac{f(x)}{g(x)}$?
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Using the chain rule, what is the derivative of the function $(f(x))^3$?
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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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Description
Test your knowledge on the derivatives of trigonometric functions and differentiation rules. This quiz covers key formulas and examples, including the power, product, quotient, and chain rules. Challenge yourself to master these essential calculus concepts!
