Calculus:Derivative of Polynomial Functions

Derivative of Polynomial Functions

=Newton Derivative of Polynomial Functions=

1. The sum rule $(f+g)'=f'+g'$
2. The Difference Rule $(f-g)'=f'-g'$
3. The Product Rule $(f*g)'=f*g'+g*f'$
4. The Quotient Rule $({f \over g})'={(gf'-fg') \over g^{2}}$

=Leibniz Derivative of Polynomial Functions=

1. The sum rule ${d(f+g) \over dx}={df \over dx}+{dg \over dx}$
2. The Difference Rule ${d(f-g) \over dx}={df \over dx}-{dg \over dx}$
3. The Product Rule ${d(fg) \over dx}'=f{dg \over dx}+g{df \over dx}$
4. The Quotient Rule ${d({f \over g}) \over dx}={g{df \over dx}-f{dg \over dx} \over g^{2}}$

Examples

Example 1

Ex1:$f'(x^{4}+2x^{2}+4x+2)$

$(f+g)'=f'+g'$

Using the sum rule we can divided in to different part $f'(x^{4}+2x^{2}+4x+2)=(x^{4})+2(x^{2})+4(x)+(2)$

so we will started to work on different part by using power rule.

$f'(x)=(x^{4})+2(x^{2})+4(x)+(2)$

$(4x^{3})+2(2x)+4$

$4x^{3}+4x+4$

Example 2

Ex2:$f'(x)=x^{4}*x^{3}$

The Product Rule $(f*g)'=f*g'+g*f'$

Using the Product Rule we can divided in to different part $f'(x^{4}*x^{3})=x^{4}*(x^{3})'+(x^{4})'*x^{3})$

$x^{4}*3x^{2}+4x^{3}*x^{3}$

$3x^{6}+4x^{6}$

$3x^{6}+4x^{6}$

$7x^{6}$

Example 3

Ex3:$f'(x)={{4x^{4}-x^{2}+10x} \over {4x^{5}-x^{3}+5x}}$

The Quotient Rule $({f \over g})'={(gf'-fg') \over g^{2}}$

$f'(x)={{(4x^{5}-x^{3}+5x)}{(4x^{4}-x^{2}+10x)'}-(4x^{4}-x^{2}+10x){(4x^{5}-x^{3}+5x)'} \over {(4x^{5}-x^{3}+5x)}^{2}}$

$f'(x)={{(4x^{5}-x^{3}+5x)}{(4x^{4}-x^{2}+10x)'}-(4x^{4}-x^{2}+10x){(4x^{5}-x^{3}+5x)'} \over {(16x^{8}-8x^{6}+80x^{5}+x^{4}-20x^{3}+100x^{2})}}$

$f'(x)={{(4x^{5}-x^{3}+5x)}{(16x^{3}-2x+10)}-(4x^{4}-x^{2}+10x){(4x^{5}-x^{3}+5x)'} \over {(16x^{8}-8x^{6}+80x^{5}+x^{4}-20x^{3}+100x^{2})}}$

$f'(x)={{(4x^{5}-x^{3}+5x)}{(16x^{3}-2x+10)}-(4x^{4}-x^{2}+10x){(20x^{4}-3x^{2}+5)} \over {(16x^{8}-8x^{6}+80x^{5}+x^{4}-20x^{3}+100x^{2})}}$

$f'(x)={{(4x^{5}-x^{3}+5x)}{(16x^{3}-2x+10)}-(4x^{4}-x^{2}+10x){(20x^{4}-3x^{2}+5)} \over {(16x^{8}-8x^{6}+80x^{5}+x^{4}-20x^{3}+100x^{2})}}$ <math>f'(x) ={ {(64x^8 - 24x^6 + 40x^5 + 82x^4 - 10x^3 - 10x^2 + 50x) - (80x^8 - 32x^6 + 200x^5 + 23x^4 - 30x^3 - 5x^2 + 50x)} \over {(16x^8-8x^6+80x^5+x^4-20x^3+100x^2)}}