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Derivative of sin(x) from the limit definition

Recall that the derivative of a function ff using the limit definition is f′(x)=lim⁡h→0f(x+h)−f(x)h.f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}. You may find the angle addition formula sin⁡(x+h)=sin⁡xcos⁡h+cos⁡xsin⁡h\sin(x+h) = \sin x \cos h + \cos x \sin h useful, along with the known limits lim⁡h→0sin⁡hh=1\lim_{h \to 0} \frac{\sin h}{h} = 1 and lim⁡h→0cos⁡h−1h=0\lim_{h \to 0} \frac{\cos h - 1}{h} = 0.

Let f(x)=sin⁡(x)f(x) = \sin(x). Using the limit definition of the derivative, f′(x)=lim⁡h→0f(x+h)−f(x)h,f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}, find a simplified expression for f′(x)f'(x).

Hint

Start by writing f(x+h)−f(x)=sin⁡(x+h)−sin⁡(x)f(x+h) - f(x) = \sin(x+h) - \sin(x), then expand sin⁡(x+h)\sin(x+h) using the angle addition formula. Group terms to separate factors of sin⁡(x)\sin(x) and cos⁡(x)\cos(x), then split the limit into two pieces and apply the two standard trigonometric limits as h→0h \to 0.

Step-by-step solution

First substitute into the difference quotient: sin⁡(x+h)−sin⁡(x)h=sin⁡xcos⁡h+cos⁡xsin⁡h−sin⁡xh.\frac{\sin(x+h) - \sin(x)}{h} = \frac{\sin x \cos h + \cos x \sin h - \sin x}{h}. Group the terms containing sin⁡x\sin x together: sin⁡x(cos⁡h−1)+cos⁡xsin⁡hh=sin⁡x⋅cos⁡h−1h+cos⁡x⋅sin⁡hh.\frac{\sin x(\cos h - 1) + \cos x \sin h}{h} = \sin x \cdot \frac{\cos h - 1}{h} + \cos x \cdot \frac{\sin h}{h}.

Now take the limit as h→0h \to 0. Since lim⁡h→0cos⁡h−1h=0\lim_{h\to 0} \frac{\cos h - 1}{h} = 0 and lim⁡h→0sin⁡hh=1\lim_{h \to 0} \frac{\sin h}{h} = 1, the expression becomes f′(x)=sin⁡x⋅0+cos⁡x⋅1=cos⁡x.f'(x) = \sin x \cdot 0 + \cos x \cdot 1 = \cos x. Thus the derivative of sin⁡(x)\sin(x) is cos⁡(x)\cos(x).

Limit value multiplying cosine x

In the derivation shown in part (a), what number multiplies cos⁡x\cos x right before taking the final limit?

Hint

Look at the form of the expression just before taking the limit. You have a product of cos⁡x\cos x and a fraction involving hh. Identify what that fraction approaches as h→0h \to 0.

Step-by-step solution

The answer is one, since the term cosine x is multiplied by the quantity sine h divided by h. As h approaches zero, this quantity approaches one. This is why the final result becomes cosine x times one, simplifying to just cosine x.

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