Derivative of `sin(x)*cos(x)`

Step-by-step differentiation

Answer

\frac{d}{d x} [sin (x) \cdot cos (x)]

cos (x)^{2} - sin (x)^{2}

Step-by-step solution

1Product Rule
$\frac{d}{d x} [sin (x) \cdot cos (x)] = \frac{d}{d x} [sin (x)] \cdot cos (x) + sin (x) \cdot \frac{d}{d x} [cos (x)]$
Let $f = sin (x)$ and $g = cos (x)$ . The product rule says $(f \cdot g)^{'} = f^{'} \cdot g + f \cdot g^{'}$ — find each factor's derivative, then assemble.
↳Find f′: Differentiate the first factor, $f = sin (x)$ :
2Derivative of sin
$\frac{d}{d x} [sin (x)] = cos (x)$
The derivative of $sin (x)$ is $cos (x)$ — a standard rule.
↳Find g′: Differentiate the second factor, $g = cos (x)$ :
3Derivative of cos
$\frac{d}{d x} [cos (x)] = - sin (x)$
The derivative of $cos (x)$ is $- sin (x)$ — a standard rule.
=Combine
$\frac{d}{d x} [sin (x) \cdot cos (x)] = cos (x) \cdot cos (x) + sin (x) \cdot - sin (x) = cos (x)^{2} - sin (x)^{2}$
With $f^{'} = cos (x)$ and $g^{'} = - sin (x)$ , substitute into $f^{'} \cdot g + f \cdot g^{'}$ and simplify.
✓Final Answer
$\frac{d}{d x} [sin (x) \cdot cos (x)] = cos (x)^{2} - sin (x)^{2}$
The first derivative simplifies to $cos (x)^{2} - sin (x)^{2}$ .

Understanding this derivative

Differentiating sin(x)·cos(x) exercises 2 distinct techniques — the product rule and the trigonometric derivatives. Problems like this one are useful practice precisely because the rules have to be combined in the right order rather than applied in isolation.

Because two expressions are multiplied together, the product rule applies: differentiate each factor in turn while keeping the other fixed, then add the results. Note that the derivative of a product is not simply the product of the derivatives. The trigonometric derivatives follow the standard cycle: sine differentiates to cosine, cosine to negative sine, and tangent to secant squared. The sign on the cosine derivative is a frequent source of errors, so it is worth double-checking.

A good way to verify a derivative by hand is to compare it against the step-by-step breakdown above — each step names the rule being applied, so you can pinpoint exactly where your own work diverges if the answers differ.

Learn the rules used here

Product & Quotient Rules Derivatives of Transcendental Functions

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Derivative of sin(x)*cos(x)

Step-by-step solution

Understanding this derivative

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Derivative of `sin(x)*cos(x)`