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Property 18

Derivative of Piecewise Function

Derivative of customed defined function can be taken in some specific points. If derivative of function on critical points is needed the continuity of function and left/right hand derivatives are analysed.

If function is continuous and right-hand derivative and left-hand derivatives are equal than function is differentiable. the derivative of function is equal to its left-hand and right-hand derivative.

If functions is non-continuous on a point then there is no derivative.

If function is continuous but right-hand derivative is different from left-hand derivative then there is no derivative on that point.

\(f(x)=\begin{cases}h(x)&x\ge a\\g(x)&x<a\end{cases}\)

a is critical point

1.
\(f(x)=\begin{cases}2x^2+6&x<2\\8x-2&x\ge2\end{cases}\)
\(\Rightarrow f'(2)=?\)
2.
\(f(x)=\begin{cases}x^2+x&x\le4\\9x-16&x>4\end{cases}\)
\(\Rightarrow f'(5)+f'(4)=?\)
3.

Derivative of function f can be taken on point 1.

\(f(x)=\begin{cases}4x+2&x>1\\2x^2+k&x\le1\end{cases}\)
\(\Rightarrow k+f'(1)=?\)
4.

Derivative of function f can be taken on point 2.

\(f(x)=\begin{cases}3x^2+2&x<2\\a&x=2\\12x+m&x>2\end{cases}\)
\(\Rightarrow m+a+f'(2)=?\)
5.

Derivative of function f can be taken on point 4.

\(f(x)=\begin{cases}6x^2+k&x<4\\t&x=4\\x^3&x>4\end{cases}\)
\(\Rightarrow f'(4)+k+t=?\)
6.

Derivative of function f can be taken on point 2.

\(f(x)=\begin{cases}3x^2-k&x>2\\m&x=2\\12x&x<2\end{cases}\)
\(\Rightarrow f'(2)+m+k=?\)
7.

Derivative of function f can be taken on point 1.

\(f(x)=\begin{cases}2x^3-k&x>1\\m&x=1\\6x+4&x<1\end{cases}\)
\(\Rightarrow 2f'(1)-m+\samefrac k2=?\)