Property 24
Geometric Interpretation of Derivative
\(m_T\); in x₀ point the drawn is tangent of slope
\(m_N\); in x₀ point the drawn is normal's slope
A function's x₀ derivative point is equal to that points drawn tangent's slope
\(\boxed{m_T=f'(x_0)}\)
in x₀ point because the drawn tangent and normal are orthogonal (right) to each other
\(m_T\cdot m_N=-1\)
\(\Rightarrow\boxed{m_N=-\samefrac1{f'(x_0)}}\)
the equation of tangent of x₀ point
\(\boxed{y-f(x_0)=f'(x_0)\cdot(x-x_0)}\)
the equation of normal of x₀ point
\(\boxed{y-f(x_0)=-\samefrac1{f'(x_0)}\cdot(x-x_0)}\)
1.
\(\Rightarrow f'(5)=?\)
2.
\(\Rightarrow f'(2)=?\)
3.
\(\Rightarrow f'(2)=?\)