Differentiating Polynomials, Composite Functions, and Products/Quotients
Differentiation is a powerful tool that allows us to find the rate of change of a function.
Differentiating Polynomials with Rational Exponents
- The power rule is a fundamental differentiation technique.
- It states that if $y = x^n$, then the derivative is:
$$ \frac{dy}{dx} = nx^{n-1} $$
The Chain Rule for Composite Functions
- When differentiating composite functions - functions nested within each other - the chain rule is used.
- If a function is of the form $f(x)=g(h(x))$, then its derivative is given by:
$$
\frac{d}{d x} f(x)=g^{\prime}(h(x)) \cdot h^{\prime}(x)
$$
Differentiating $f(x)=e^{\left(x^2+2\right)}$ requires applying the chain rule, yielding $f^{\prime}(x)=$ $e^{\left(x^2+2\right)} \cdot 2 x$
The Product Rule for Differentiating Products of Functions
- The product rule is used when differentiating the product of two functions.
- If $y = u(x) \cdot v(x)$, then:
$$ \frac{dy}{dx} = u'(x) \cdot v(x) + u(x) \cdot v'(x) $$
A common mistake is to forget one of the terms in the product rule. Always remember: differentiate one function at a time while keeping the other unchanged.
The Quotient Rule for Differentiating Quotients of Functions
- The quotient rule is used when differentiating the division of two functions.
- If $y = \frac{u(x)}{v(x)}$, then:
$$ \frac{dy}{dx} = \frac{v(x) \cdot u'(x) - u(x) \cdot v'(x)}{(v(x))^2} $$
When using the quotient rule, remember the order: the derivative of the numerator comes first, followed by the derivative of the denominator.
Combining Rules: A Step-by-Step Approach
Differentiating complex functions often requires combining multiple rules.
- Identify the structure: Is it a product, quotient, or composite function?
- Choose the appropriate rule: Use the product rule, quotient rule, or chain rule as needed.
- Differentiate systematically: Break the problem into smaller parts and differentiate each part.
- Simplify the result: Combine and simplify terms for a clean final answer.
Try differentiating $y = \frac{\sin(x)}{x^2 + 1}$. Which rule(s) will you use?
- How does the ability to differentiate complex functions reflect the interconnectedness of mathematical rules?
- Can you think of other areas where combining multiple techniques is essential?
