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Find the derivative of $53\sin\left(x\right)$ 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 $53\sin\left(x\right)$. Substituting $f(x+h)$ and $f(x)$ on the limit, we get
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$\lim_{h\to0}\left(\frac{53\sin\left(x+h\right)-53\sin\left(x\right)}{h}\right)$
Learn how to solve definition of derivative problems step by step online. Find the derivative of 53sin(x) using the definition. Find the derivative of 53\sin\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 53\sin\left(x\right). Substituting f(x+h) and f(x) on the limit, we get. Using the sine of a sum formula: \sin(\alpha\pm\beta)=\sin(\alpha)\cos(\beta)\pm\cos(\alpha)\sin(\beta), where angle \alpha equals x, and angle \beta equals h. Multiply the single term 53 by each term of the polynomial \left(\sin\left(x\right)\cos\left(h\right)+\cos\left(x\right)\sin\left(h\right)\right). Factor the polynomial 53\sin\left(x\right)\cos\left(h\right)+53\cos\left(x\right)\sin\left(h\right)-53\sin\left(x\right) by it's greatest common factor (GCF): 53.