d/dx (sin x)
cos x
To find the derivative of sin x, we will use the definition of the derivative:
d/dx (sin x) = lim(h->0) [(sin(x + h) – sin x)/h]
We can then use the trigonometric identity sin (a + b) = sin a cos b + cos a sin b to write:
d/dx (sin x) = lim(h->0) [(sin x cos h + cos x sin h – sin x)/h]
We can then factor out sin x from the numerator and cancel it out with the denominator:
d/dx (sin x) = lim(h->0) [(sin x(cos h – 1)/h) + (cos x sin h)/h]
Taking the limits as h approaches 0, the first term simplifies to -sin x and the second term simplifies to cos x:
d/dx (sin x) = -sin x + cos x
Therefore, the derivative of sin x is cos x.
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