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Find the derivative of $5\sin\left(x\right)$ 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 $5\sin\left(x\right)$. Substituting $f(x+h)$ and $f(x)$ on the limit, we get
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$\lim_{h\to0}\left(\frac{5\sin\left(x+h\right)-5\sin\left(x\right)}{h}\right)$
Learn how to solve problems step by step online. Find the derivative of 5sin(x) using the definition. Find the derivative of 5\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 5\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 5 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 5\sin\left(x\right)\cos\left(h\right)+5\cos\left(x\right)\sin\left(h\right)-5\sin\left(x\right) by it's greatest common factor (GCF): 5.