Respuestas
Yo te ayudo
1 ) f'(x)= 8 -10x
2) f'(x) = -6x /( 1 + x²)
3) f'(x) = 3/√6x+1
Para derivar por definicion se procede a aplicar la siguiente fórmula :
f'(x) = Lim h→0 ( f( x + h ) - f(x) ) /h
1) f(x) = 2+8x-5x²
f'(x)= Lim h→0 [ ( 2 + 8(x +h ) - 5* ( x+h)² ) - ( 2 + 8x -5x² ) ]/h
f'(x)= Lim h→ 0 ( 2 + 8x +8h - 5x² -10xh -5h²-2-8x +5x² )/h
f'(x)= Lim h→0 ( 8h -10xh-5h² )/h
f'(x) = Lim h →0 h( 8 -10x -5h)/h
f'(x) = Lim h→0 ( 8 -10x -5h ) = 8 -10x
2) f(x )= 3/ 1+x²
f'(x) = lim h→0 [ 3/( 1+ ( x+h)² ) - 3/( 1+ x² ) ]/h
= lim h→0 [ 3/ ( 1+x²+2xh+h²) - 3/(1+x²)]/h
= lim h→0 ( 3+3x²-3-3x²-6xh-3h²)/h*(1+x²+2xh+h²)(1+x²)
= lim h→0 ( -6xh -3h²)/(h(1+x²+2xh+h²)(1+x²) )
= lim h→0 ( -6x -3h )/( 1+x² +2xh +h²)(1+x²)
f'(x) = -6x /( 1 + x²)²
3) f(x) = √6x+1
f'(x) = lim h→0 (√6*( x+h)+1 - √6x+1 )/h
= lim h→0 ( √6x+6h+1 - √6x+1 )/h
= lim h→0 ( √6x +6h+1 - √6x+1 )*( √6x+6h+1 + √6x+1 )/h*(√6x+6h+1 +√6x+1 )
= lim h→0 [ ( √6x+6h+1 )²- ( √6x+1 )²]/h*(√6x+6h+1 +√6x+1 )
= lim h→0 ( 6x+6h+1-6x-1 )/h* ( √6x+6h+1 +√6x+1 )
= lim h→0 6h/h*(√6x+6h+1 + √6x+1 )
= lim h→0 6/( √6x+6h+1 + √6x+1 )
f'(x)= 6/(√6x +1 +√6x +1 )
f'(x) =6/2√6x+1
f'(x) = 3/√6x+1