Questions and Answers
What is the derivative of $e^x$?
The derivative of $ an(x)$ is $ rac{1}{ ext{cos}^2(x)}$.
False
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 _______.
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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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Description
Explore the essential differentiation formulas in calculus through this quiz. Understand the derivatives of basic, logarithmic, exponential, and trigonometric functions. Test your knowledge on how to calculate rates of change and slopes for various functions.
