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Find the derivative of $\cos\left(x\right)$ using the definition. Apply the definition of the derivative: $\displaystyle f'(x)=\lim_{h\to0}\frac{f(x+h)-f(x)}{h}$. The function $f(x)$ is the function we want to differentiate, which is $\cos\left(x\right)$. Substituting $f(x+h)$ and $f(x)$ on the limit, we get
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$\lim_{h\to0}\left(\frac{\cos\left(x+h\right)-\cos\left(x\right)}{h}\right)$
Learn how to solve definition of derivative problems step by step online. Find the derivative of cos(x) using the definition. Find the derivative of \cos\left(x\right) using the definition. Apply the definition of the derivative: \displaystyle f'(x)=\lim_{h\to0}\frac{f(x+h)-f(x)}{h}. The function f(x) is the function we want to differentiate, which is \cos\left(x\right). Substituting f(x+h) and f(x) on the limit, we get. Using the cosine of a sum formula: \cos(\alpha\pm\beta)=\cos(\alpha)\cos(\beta)\mp\sin(\alpha)\sin(\beta), where angle \alpha equals x, and angle \beta equals h. Factoring by \cos\left(x\right). Expand the fraction \frac{\cos\left(x\right)\left(\cos\left(h\right)-1\right)-\sin\left(x\right)\sin\left(h\right)}{h} into 2 simpler fractions with common denominator h.