Derivadas usando a definição de limite

Derivadas usando a definição de limite

Os problemas seguintes requerem o uso da definição de derivada por limite, logo: 1) Use a definição de derivada via limite para calcular a derivada f'(x), para:

1.a) f^' (x)=lim┬(∆x→0)⁡〖(f(x+∆x)-f(x))/∆x〗 f^' (x)=lim┬(∆x→0)⁡〖((1/2(x+∆x)-3/5)-(1/2 x-3/5))/∆x〗 f^' (x)=lim┬(∆x→0)⁡〖(x/2+∆x/2-3/5-x/2+3/5)/∆x〗 f^' (x)=lim┬(∆x→0)⁡〖(∆x/2)/∆x〗 f^' (x)=lim┬(∆x→0)⁡〖∆x/2*1/∆x〗=1/2

1.b) f^' (x)=lim┬(∆x→0)⁡〖(f(x+∆x)-f(x))/∆x〗 f^' (x)=lim┬(∆x→0)⁡〖((5(x+∆x)^2-3(x+∆x)+7)-(5x^2-3x+7))/∆x〗 f^' (x)=lim┬(∆x→0)⁡〖((5*(x^2+2*x*∆x+〖∆x〗^2)-3x-3∆x+7)-5x^2+3x-7)/∆x〗 f^' (x)=lim┬(∆x→0)⁡〖(5x^2+10*x*∆x+5〖∆x〗^2-3x-3∆x+7-5x^2+3x-7)/∆x〗 f^' (x)=lim┬(∆x→0)⁡〖(10*x*∆x+5〖∆x〗^2-3∆x)/∆x〗 f^' (x)=lim┬(∆x→0)⁡〖(∆x(10x+5∆x-3))/∆x〗 f^' (x)=lim┬(∆x→0)⁡〖10x+5∆x-3〗 f^' (x)=lim┬(∆x→0)⁡〖10x+5*0-3〗=10x-3

1.c) f^' (x)=lim┬(∆x→0)⁡〖(f(x+∆x)-f(x))/∆x〗 f^' (x)=lim┬(∆x→0)⁡〖((4-√((x+∆x)+3))-(4-√(x+3)))/∆x〗 f^' (x)=lim┬(∆x→0)⁡〖(√(x+3)-√((x+∆x)+3))/∆x〗*(√(x+3)+√((x+∆x)+3))/(√(x+3)+√((x+∆x)+3)) f^' (x)=lim┬(∆x→0)⁡〖((x+3)-(x+∆x+3))/(∆x(√(x+3)+√((x+∆x)+3)))〗 f^' (x)=lim┬(∆x→0)⁡〖(-∆x)/(∆x(√(x+3)+√((x+∆x)+3)))〗 f^' (x)=lim┬(∆x→0)⁡〖(-1)/(√(x+3)+√((x+∆x)+3))〗 f^' (x)=lim┬(∆x→0)⁡〖(-1)/(√(x+3)+√(x+3))〗 f^' (x)=lim┬(∆x→0)⁡〖(-1)/(2√(x+3))〗

1.d) f^' (x)=lim┬(∆x→0)⁡〖(f(x+∆x)-f(x))/∆x〗 f^' (x)=lim┬(∆x→0)⁡〖((((x+∆x)+1)/(2-(x+∆x)))-((x+1)/(2-x)))/∆x〗 f^' (x)=lim┬(∆x→0)⁡〖(((x+∆x+1)/(2-x-∆x))-((x+1)/(2-x)))/∆x〗 f^' (x)=lim┬(∆x→0)⁡〖(((x+∆x+1)/(2-x-∆x))-((x+1)/(2-x)))/∆x〗 f^' (x)=lim┬(∆x→0)⁡〖((2x+2∆x+2-x^2-x∆x-x-(2x+2-x^2-x-x∆x-∆x))/((2-x-∆x)*(2-x)))/∆x〗 f^' (x)=lim┬(∆x→0)⁡〖((2x+2∆x+2-x^2-x∆x-x-2x-2+x^2+x+x∆x+∆x)/((2-x-∆x)*(2-x)))/∆x〗 f^' (x)=lim┬(∆x→0)⁡〖(3∆x/((2-x-∆x)*(2-x)))/(∆x/1)〗 f^' (x)=lim┬(∆x→0)⁡〖3∆x/((2-x-∆x)*(2-x))*1/∆x〗 f^' (x)=lim┬(∆x→0)⁡〖3/((2-x)*(2-x))〗 f^' (x)=lim┬(∆x→0)⁡〖3/(2-x)^2 〗

1.e) f^' (x)=lim┬(∆x→0)⁡〖(f(x+∆x)-f(x))/∆x〗 f^' (x)=lim┬(∆x→0)⁡〖(〖(x+∆x)〗^(2/3)-x^(2/3))/∆x〗
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