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Derivative of sin x

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Derivative f’ of the function f(x)=sinx is: f’(x) = cos x for any value of x.

Derivative of sin x

Derivative $f’$ of the function $f(x)=\sin x$ is:

$$ \forall x \in ]-\infty, +\infty[ , f’(x) = \cos x $$

Proof/Demonstration

$$ \begin{aligned} \frac{\sin (x+h)-\sin x}{h}&= \frac{\sin (x) \cos (h)+\cos (x) \sin (h)-\sin x}{h} \\ \frac{\sin (x+h)-\sin x}{h} &=\frac{\sin h}{h} \times \cos x+\sin x \times\frac{\cos h}{h} -\sin x \times \frac{1}{h}\\ \frac{\sin (x+h)-\sin x}{h} &=\frac{\sin h}{h} \times \cos x+\sin x \times \frac{\cos h-1}{h} \end{aligned} $$

We have

$$ \begin{aligned} \frac{\cos h-1}{h} &=\frac{(\cos h-1)(\cos h+1)}{h(\cos h+1)} \\ &=\frac{\cos ^{2} h-1}{h(\cos h+1)} \\ &=\frac{-\sin ^{2} h}{h(\cos h+1)} \\ &=\frac{\sin h}{h} \times \frac{-\sin h}{\cos h+1} \\ \end{aligned} $$

Then

$$ \lim _{h \rightarrow 0} \frac{\cos h-1}{h}=0 $$

because

$$ \lim _{h \rightarrow 0} \frac{\sin h}{h}=1 $$
This equality has been proved in https://www.math-linux.com/mathematics/limits/article/proof-of-limit-of-sin-x-x-1-as-x-approaches-0

Now

$$ \begin{aligned} \lim _{h \rightarrow 0} \frac{\sin (x+h)-\sin x}{h} &=\lim _{h \rightarrow 0} \frac{\sin h}{h} \times \cos x+\sin x \times \lim _{h \rightarrow 0} \frac{\cos h-1}{h}\\ &=1\times \cos x+\sin x \times 0 \end{aligned} $$

We conclude:

$$ \lim _{h \rightarrow 0} \frac{\sin (x+h)-\sin x}{h}=\cos x $$

Also in this section

  1. Derivative of ln x
  2. Derivative of 1/x
  3. Derivative of cos x
  4. Derivative of tan x
  5. Derivative of exp x, e^x
  6. Derivative of sin x
  7. Derivative of inverse functions
  8. Chain rule proof - derivative of a composite function