Let g(x) = cos(x). Find g'(x), and then evaluate g'left(π/3right).

Derivative of cosine evaluated at a point

Consider the function $g(x) = \cos(x)$ .

Let $g(x) = \cos(x)$ . Find $g'(x)$ , and then evaluate $g'\left(\frac{\pi}{3}\right)$ .

Hint

Recall the memorized derivative rule for $\cos(x)$ , then substitute $x = \pi/3$ into the resulting expression and evaluate the sine of that angle.

Step-by-step solution

The derivative of $\cos(x)$ is a standard trigonometric derivative that should be memorized: $\dfrac{d}{dx}[\cos(x)] = -\sin(x)$ . So $g'(x) = -\sin(x)$ .

Now substitute $x = \pi/3$ : $g'\left(\frac{\pi}{3}\right) = -\sin\left(\frac{\pi}{3}\right)$ . Since $\sin(\pi/3) = \frac{\sqrt{3}}{2}$ , we get $g'\left(\frac{\pi}{3}\right) = -\frac{\sqrt{3}}{2}$ .

Derivative of cosine evaluated at a point

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