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Questions and Answers
What is the derivative of $e^x$?
What is the derivative of $e^x$?
- $x^e$
- $e^x$ (correct)
- $xe^x$
- $ rac{1}{e^x}$
The derivative of $ an(x)$ is $ rac{1}{ ext{cos}^2(x)}$.
The derivative of $ an(x)$ is $ rac{1}{ ext{cos}^2(x)}$.
False (B)
What is the formula for the derivative of the natural logarithm function $ rac{d}{dx}( ext{ln}(x))$?
What is the formula for the derivative of the natural logarithm function $ rac{d}{dx}( ext{ln}(x))$?
frac{1}{x}
The derivative of $ ext{sin}(x)$ is _______.
The derivative of $ ext{sin}(x)$ is _______.
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Match the function with its derivative:
Match the function with its derivative:
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Study Notes
Differentiation Formulas
- Differentiation is a fundamental operation in calculus, allowing the calculation of rates of change and slopes of functions.
Derivatives of Basic Functions
- The derivative of a constant function is zero.
- For ( f(x) = x^n ), the derivative is ( f'(x) = nx^{n-1} ).
- Common derivatives include:
- ( f(x) = e^x ) leads to ( f'(x) = e^x )
- ( f(x) = a^x ) results in ( f'(x) = a^x \ln(a) )
Derivatives of Logarithmic and Exponential Functions
- For natural logarithm ( f(x) = \ln(x) ), the derivative is ( f'(x) = \frac{1}{x} ).
- The derivative for logarithm base ( a ) is ( f(x) = \log_a(x) ) which gives ( f'(x) = \frac{1}{x \ln(a)} ).
- The derivative of the exponential function ( f(x) = e^{g(x)} ) is ( f'(x) = e^{g(x)} g'(x) ).
Derivatives of Trigonometric Functions
- Key trigonometric derivatives:
- ( f(x) = \sin(x) ) leads to ( f'(x) = \cos(x) )
- ( f(x) = \cos(x) ) results in ( f'(x) = -\sin(x) )
- ( f(x) = \tan(x) ) leads to ( f'(x) = \sec^2(x) )
Derivatives of Inverse Trigonometric Functions
- For inverse functions:
- ( f(x) = \arcsin(x) ) gives ( f'(x) = \frac{1}{\sqrt{1-x^2}} )
- ( f(x) = \arccos(x) ) results in ( f'(x) = -\frac{1}{\sqrt{1-x^2}} )
- ( f(x) = \arctan(x) ) leads to ( f'(x) = \frac{1}{1+x^2} )
Differentiation Rules
- Product Rule: For ( f(x) = u(x)v(x) ), ( f'(x) = u'v + uv' ).
- Quotient Rule: For ( f(x) = \frac{u(x)}{v(x)} ), ( f'(x) = \frac{u'v - uv'}{v^2} ).
- Chain Rule: For composite functions, ( f(g(x)) ), the derivative is ( f'(g(x)) \cdot g'(x) ).
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