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The Product Rule is essential when differentiating the product of two functions. For two functions \( f(x) \) and \( g(x) \), the derivative of their product is:
\((f \cdot g)' = f' \cdot g + f \cdot g'\)
Example: Let \( f(x) = 3x^2 \) and \( g(x) = 5x \), then their derivative is \( (3x^2)' \cdot 5x + 3x^2 \cdot (5x)' \).
Question: What is the derivative of \( (2x^3) \cdot (4x) \)?
The Quotient Rule helps us find the derivative of a ratio of two functions. For \( f(x) / g(x) \), the derivative is:
\((\frac{f}{g})' = \frac{f' \cdot g - f \cdot g'}{g^2}\)
Example: If \( f(x) = x^2 \) and \( g(x) = x+1 \), then the derivative is \( \frac{(2x)(x+1) - (x^2)(1)}{(x+1)^2} \).
Question: What is the derivative of \( \frac{3x^2}{x+2} \)?
The second derivative represents the rate of change of the rate of change of a function. It provides insights into the function's concavity.
Example: If \( f(x) = x^3 \), then \( f'(x) = 3x^2 \) and \( f''(x) = 6x \).
Partial derivatives apply to functions of multiple variables and involve differentiating one variable while keeping others constant. For \( f(x, y) \), the partial derivative with respect to \( x \) is \( \frac{\partial f}{\partial x} \).
Example: For \( f(x, y) = x^2 y + y^3 \), the partial derivative with respect to \( x \) is \( 2xy \).
The Chain Rule helps us differentiate composite functions. If \( y = f(g(x)) \), then:
\( y' = f'(g(x)) \cdot g'(x) \)
Example: If \( f(x) = (3x^2 + 2)^2 \), find the derivative:
\( y' = 2(3x^2 + 2) \cdot (6x) \)
Integration is the reverse of differentiation. It helps calculate areas under curves and solve differential equations. For example:
\(\int x^2 dx = \frac{x^3}{3} + C\)
Differentiation is a fundamental concept in calculus, and the following rules make it easier to compute derivatives efficiently:
Answer the following questions to test your understanding. Feedback with calculations will be provided!
Visualize a function \( f(x) \), its tangent at a specified point, and inflection points. Explore interactive graphs and detailed explanations.
Challenge your understanding of Grade 12 calculus concepts! Solve each question, then click the "Show Solution" button to view the formula and solution.
Question: Differentiate \( f(x) = 3x^2 + 5x - 4 \).
Question: Find the derivative of \( g(x) = \sin(x) \).
Question: Evaluate the integral \( \int (4x^3 - 2x)dx \).
Question: Determine the critical points of \( h(x) = x^3 - 3x^2 + 4 \).
Question: Compute the limit: \( \lim_{x \to \infty} \frac{2x^2 + 3x}{5x^2 + 7} \).
Challenge your understanding of Grade 12 calculus concepts! Solve each question, then click the "Show Solution" button to view the formula and solution.
Question: Plot the graph of \( f(x) = x^2 - 4 \) for \( x \in [-3, 3] \).
Question: Plot the graph of \( g(x) = \sin(x) \) for \( x \in [0, 2\pi] \).
Question: Plot the graph of \( h(x) = e^x \) for \( x \in [-2, 2] \).
Question: Plot the graph of \( k(x) = \ln(x) \) for \( x \in [0.1, 5] \).
Question: Plot the graph of \( m(x) = \cos(x) \) for \( x \in [0, 2\pi] \).
Definition: A logarithm is the inverse of exponentiation. It answers the question: "To what exponent must the base be raised, to produce a given number?"
Mathematically, the logarithm of a number \(x\) with respect to a base \(b\) is defined as:
\[ \log_b(x) = y \iff b^y = x \]
where:
The logarithm of zero, \( \log_b(0) \), is **undefined**. This is because there is no real exponent that can make a positive base equal zero. Specifically, for any positive base \(b\):
\[ b^y = 0 \]
has no solution in real numbers.
As the input approaches zero from the positive side, the logarithmic function tends toward negative infinity. Formally:
\[ \lim_{{x \to 0^+}} \log_b(x) = -\infty \]
This behavior can be visualized in the graph of the logarithmic function, which descends infinitely as it approaches zero.
Mathematical Justification:
Logarithms are widely used in various fields of science and mathematics:
Conclusion: The logarithm of zero is undefined because no exponent can make a positive base equal zero. This fundamental property of logarithms forms the basis for their behavior and limitations in mathematical functions.