Learn How To Find The Derivative Of Sinx With Trigonometric Identities.

derivative of sinx

cosx

The derivative of sinx is cosx.

To derive this result, we can use the definition of derivatives and the trigonometric identities.

We begin with the definition of the derivative:

f'(x) = lim(h -> 0) [f(x+h) – f(x)]/h

Let f(x) = sinx. Then:

f'(x) = lim(h -> 0) [sin(x+h) – sin(x)]/h

We can use the identity:

sin(a+b) = sin(a)cos(b) + cos(a)sin(b)

to rewrite the numerator:

sin(x+h) – sin(x) = sin(x)cos(h) + cos(x)sin(h) – sin(x)
= cos(h)sin(x)

We substitute this into the derivative formula:

f'(x) = lim(h -> 0) [cos(h)sin(x)]/h

Using the limit definition of the cosine function:

lim(h -> 0) [cos(h) – 1]/h = 0

We can rewrite the derivative as:

f'(x) = lim(h -> 0) [cos(h)sin(x)]/h
= lim(h -> 0) cos(h) * lim(h -> 0) sin(x)/h
= cos(0) * sin(x)
= cos(x)

Thus:

(f(x))’ = cos(x)

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