Производная от Sin ax

Какова Производная от Sin ax ?

Производная от sin ax равна a.cos ax.

$(sinax{)}^{\prime }=a.cosax$

$\frac{d}{dx}(sinax)=a.cosax$

Доказательство Производной от Sin ax

Метод 1

$f^{\prime }(x)=\underset{h\to 0}{\lim }\frac{f(x+h)-f(x)}{h}$

$(sinax{)}^{\prime }=\underset{h\to 0}{\lim }\frac{sina(x+h)-sinax}{h}$

$(sinax{)}^{\prime }=\underset{h\to 0}{\lim }\frac{sin(ax+ah)-sinax}{h}$

$\boxed{sin(p+q)=sinp.cosq+cosp.sinq}$

$(sinax{)}^{\prime }=\underset{h\to 0}{\lim }\frac{sinax.cosah+cosax.sinah-sinax}{h}$

$(sinax{)}^{\prime }=\underset{h\to 0}{\lim }\frac{sinax.cosah-sinax+cosax.sinah}{h}$

$(sinax{)}^{\prime }=\underset{h\to 0}{\lim }\frac{sinax.(cosah-1)+cosax.sinah}{h}$

$(sinax{)}^{\prime }=\underset{h\to 0}{\lim }\frac{a.[sinax.(cosah-1)+cosax.sinah]}{a.h}$

$(sinax{)}^{\prime }=a.\underset{h\to 0}{\lim }\frac{sinax.(cosah-1)+cosax.sinah}{ah}$

$(sinax{)}^{\prime }=a.\underset{h\to 0}{\lim }[\frac{sinax.(cosah-1)}{ah}+\frac{cosax.sinah}{ah}]$

$(sinax{)}^{\prime }=a.\underset{h\to 0}{\lim }\frac{sinax.(cosah-1)}{ah}+a.\underset{h\to 0}{\lim }\frac{cosax.sinah}{ah}$

$(sinax{)}^{\prime }=a.sinax.\underset{h\to 0}{\lim }\frac{cosah-1}{ah}+a.cosax.\underset{h\to 0}{\lim }\frac{sinah}{ah}$

$h\to 0(ah=h)$

$(sinax{)}^{\prime }=a.sinax.\underset{h\to 0}{\lim }\frac{cosh-1}{h}+a.cosax.\underset{h\to 0}{\lim }\frac{sinh}{h}$

$\boxed{\underset{t\to 0}{\lim }\frac{sint}{t}=1}$ $\boxed{\underset{t\to 0}{\lim }\frac{cost-1}{t}=0}$

$(sinax{)}^{\prime }=a.sinax.0+a.cosax.1$

$(sinax{)}^{\prime }=0+a.cosax$

$(sinax{)}^{\prime }=a.cosax$

Метод 2

$f^{\prime }(x)=\underset{h\to 0}{\lim }\frac{f(x+h)-f(x)}{h}$

$(sinax{)}^{\prime }=\underset{h\to 0}{\lim }\frac{sina(x+h)-sinax}{h}$

$(sinax{)}^{\prime }=\underset{h\to 0}{\lim }\frac{sin(ax+ah)-sinax}{h}$

$\boxed{sinp-sinq=2.sin(\frac{p-q}{2}).cos(\frac{p+q}{2})}$

$(sinax{)}^{\prime }=\underset{h\to 0}{\lim }\frac{2.sin\frac{\cancel{ax}+ah-\cancel{ax}}{2}.cos\frac{ax+ah+ax}{2}}{h}$

$(sinax{)}^{\prime }=\underset{h\to 0}{\lim }\frac{2.sin\frac{ah}{2}.cos\frac{2ax+ah}{2}}{h}$

$(sinax{)}^{\prime }=\underset{h\to 0}{\lim }\frac{2.sin\frac{ah}{2}.cos\frac{\cancel{2}.(ax+\frac{ah}{2})}{\cancel{2}}}{h}$

$(sinax{)}^{\prime }=\underset{h\to 0}{\lim }\frac{2.sin\frac{ah}{2}.cos(ax+\frac{ah}{2})}{h}$

$(sinax{)}^{\prime }=\underset{h\to 0}{\lim }\frac{sin\frac{ah}{2}.cos(ax+\frac{ah}{2})}{\frac{1}{2}.h}$

$(sinax{)}^{\prime }=\underset{h\to 0}{\lim }\frac{sin\frac{ah}{2}.cos(ax+\frac{ah}{2})}{\frac{h}{2}}$

$(sinax{)}^{\prime }=\underset{h\to 0}{\lim }\frac{a.sin\frac{ah}{2}.cos(ax+\frac{ah}{2})}{a.\frac{h}{2}}$

$(sinax{)}^{\prime }=a.\underset{h\to 0}{\lim }\frac{sin\frac{ah}{2}.cos(ax+\frac{ah}{2})}{\frac{ah}{2}}$

$(sinax{)}^{\prime }=a.\underset{h\to 0}{\lim }[\frac{sin\frac{ah}{2}}{\frac{ah}{2}}.cos(ax+\frac{ah}{2})]$

$(sinax{)}^{\prime }=a.\underset{h\to 0}{\lim }\frac{sin\frac{ah}{2}}{\frac{ah}{2}}.\underset{h\to 0}{\lim }cos(ax+\frac{ah}{2})$

$h\to 0(\frac{ah}{2}=h)$

$(sinax{)}^{\prime }=a.\underset{h\to 0}{\lim }\frac{sinh}{h}.\underset{h\to 0}{\lim }cos(ax+h)$

$(sinax{)}^{\prime }=a.1.cos(ax+0)$

$(sinax{)}^{\prime }=a.1.cosax$

$(sinax{)}^{\prime }=a.cosax$

Метод 3

$\boxed{sinax=ax-\frac{(ax{)}^3}{3!}+\frac{(ax{)}^5}{5!}-\frac{(ax{)}^7}{7!}+\frac{(ax{)}^9}{9!}-...}$

$\boxed{cosax=1-\frac{(ax{)}^2}{2!}+\frac{(ax{)}^4}{4!}-\frac{(ax{)}^6}{6!}+\frac{(ax{)}^8}{8!}-...}$

$sinax=ax-\frac{(ax{)}^3}{3!}+\frac{(ax{)}^5}{5!}-\frac{(ax{)}^7}{7!}+\frac{(ax{)}^9}{9!}-...$

$(sinax{)}^{\prime }=[ax-\frac{(ax{)}^3}{3!}+\frac{(ax{)}^5}{5!}-\frac{(ax{)}^7}{7!}+\frac{(ax{)}^9}{9!}-...{]}^{\prime }$

$(sinax{)}^{\prime }=(ax{)}^{\prime }-[\frac{(ax{)}^3}{3!}{]}^{\prime }+[\frac{(ax{)}^5}{5!}{]}^{\prime }-[\frac{(ax{)}^7}{7!}{]}^{\prime }+[\frac{(ax{)}^9}{9!}{]}^{\prime }-...$

$(sinax{)}^{\prime }=a-\frac{3.(ax{)}^2.(ax{)}^{\prime }}{3!}+\frac{5.(ax{)}^4.(ax{)}^{\prime }}{5!}-\frac{7.(ax{)}^6.(ax{)}^{\prime }}{7!}+\frac{9.(ax{)}^8.(ax{)}^{\prime }}{9!}-...$

$(sinax{)}^{\prime }=a-\frac{3.(ax{)}^2.a}{3!}+\frac{5.(ax{)}^4.a}{5!}-\frac{7.(ax{)}^6.a}{7!}+\frac{9.(ax{)}^8.a}{9!}-...$

$(sinax{)}^{\prime }=a-\frac{\cancel{3}.(ax{)}^2.a}{\cancel{3}.2!}+\frac{\cancel{5}.(ax{)}^4.a}{\cancel{5}.4!}-\frac{\cancel{7}.(ax{)}^6.a}{\cancel{7}.6!}+\frac{\cancel{9}.(ax{)}^8.a}{\cancel{9}.8!}-...$

$(sinax{)}^{\prime }=a-\frac{(ax{)}^2.a}{2!}+\frac{(ax{)}^4.a}{4!}-\frac{(ax{)}^6.a}{6!}+\frac{(ax{)}^8.a}{8!}-...$

$(sinax{)}^{\prime }=a.[1-\frac{(ax{)}^2}{2!}+\frac{(ax{)}^4}{4!}-\frac{(ax{)}^6}{6!}+\frac{(ax{)}^8}{8!}-...]$

$(sinax{)}^{\prime }=a.cosax$