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This guide explores the derivatives of sine and cosine functions through graphical representation and algebraic definition. We can visualize the slope of these functions by connecting points on their curves, resulting in cosine and sine curves respectively. The derivative of y = sin(x) is proven to be y’ = cos(x) using the definition of derivatives. Additionally, we discuss the product and quotient rules for derivatives, emphasizing that they are not just simple derivatives of products or quotients. Enhance your understanding of these fundamental concepts in calculus.
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slope Consider the function We could make a graph of the slope: Now we connect the dots! The resulting curve is a cosine curve.
slope We can do the same thing for The resulting curve is a sine curve that has been reflected about the x-axis.
Derivative of y=sinx • Use the definition of the derivative To prove the derivative of y=sinx is y’=cosx.
Derivative of y=sinx Shortcut: y’=cosx The proof of the d(cosx) = -sinx is almost identical
product rule: Notice that this is not just the product of two derivatives. This is sometimes memorized as:
