How do I us the Limit definition of derivative on f(x)=cos⁡(x)?

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blogspainmarax6qi · Accepted Answer

We know that the formula for the limit definition of the derivative is:

lim_{△ x \to 0} \frac{f (x + △ x) - f (x)}{△ x}

So let's apply it:

lim_{△ x \to 0} \frac{\cos (x + △ x) - \cos x}{△ x}

= lim_{△ x \to 0} \frac{\cos x \cdot \cos △ x - \sin x \cdot \sin △ x - \cos x}{△ x}

= lim_{△ x \to 0} \frac{\cos x (\cos △ x - 1) - \sin x \cdot \sin △ x}{△ x}

= lim_{△ x \to 0} (\cos x \cdot \frac{\cos △ x - 1}{△ x} - \sin x \cdot \frac{\sin △ x}{△ x})

= (1) (0) - (\sin x) (1)

= - \sin x

mortaraskxnl · Accepted Answer

Find the derivative of cos⁡(x) using the definition. Apply the definition of the derivative: f′(x)=limh→0f(x+h)−f(x)h. The function f(x) is the function we want to differentiate, which is cos⁡(x).Substituting f(x+h) and f(x) on the limitlimh→0(cos⁡(x+h)−cos⁡(x)h)=−sin⁡(x)

How do I us the Limit definition of

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