Differentiation

Differentiation (review) §

• differentiation
• second derivative test. $f^{\prime \prime }(x)=\frac{d^{2}y}{d{x}^2}$

The Product Rule §

• The product rules allows you to determine the derivative of the product of functions as follows: $\text{If }f(x)g(x)\text{, then }{y}^{\prime }=g(x)f^{\prime }(x)+f(x)g^{\prime }(x)$
• Alternative using Leibniz notation: $\text{If }y=uv\text{, then }\frac{dy}{dx}=v\frac{du}{dx}+u\frac{dv}{dx}$

Quotient Rule §

• Let $y=\frac{u}{v}$, the differentiation of $y$ $\frac{dy}{dx}$ is $=\frac{v{u}^{\prime }-u{v}^{\prime }}{v^2}$

The Chain Rule §

• If $y=f(g(x)\text{, the derivative }\frac{dy}{dx}\text{ can be expressed as }{f}^{\prime }(g(x))g^{\prime }(x)$
• If $y=(u{)}^n$, the derivative $\frac{dy}{dx}$ can be expressed as $n\times (u{)}^{n-1}\times u^{\prime }$

Rates of Change §

• Differentiation can be used to solve problems involving rate of change.
  • Important to know:
    • quantity you are differentiating
    • quantity you are differentiating with respect to,
  • t <= 10
  • 4/3*pi*(10-t)
  • -4*pi*(10-t)