Mathematics ACT Aspire Test 2025 – 400 Free Practice Questions to Pass the Exam

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Question: 1 / 400

Which of the following is the derivative of sin(x)?

cos(x)

The derivative of the sine function, sin(x), is a fundamental result in calculus, derived from the definition of the derivative and the properties of trigonometric functions. When we differentiate sin(x), we are essentially measuring the rate at which sin(x) changes with respect to changes in x.

Using the limit definition of the derivative, we have:

\[

\frac{d}{dx} \sin(x) = \lim_{h \to 0} \frac{\sin(x+h) - \sin(x)}{h}

\]

Applying the trigonometric identity for the sine of a sum, we rewrite sin(x + h) as:

\[

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

\]

Substituting this back into the limit gives:

\[

\frac{d}{dx} \sin(x) = \lim_{h \to 0} \frac{(\sin(x)\cos(h) + \cos(x)\sin(h)) - \sin(x)}{h}

\]

This simplifies to:

\[

\frac{d}{dx} \sin(x) = \lim_{h \to 0} \left( \frac{\

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sin(x)

-sin(x)

-cos(x)

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