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:
$$ \frac{dy}{dx} = u'(x) \cdot v(x) + u(x) \cdot v'(x) $$
