Derivatives of Trigonometric Functions

Derivatives of Sine, Cosine, and Tangent

Some common functions that appear in equations are the basic trigonometric functions. The following three theorems will establish their derivatives.

 Theorem 1: If $f(x) = \sin x$ then $\frac{d}{dx} f(x) = \cos x$.
• Proof: Let $f(x) = \sin x$. From the definition of a derivative it follows that:
(1)
\begin{align} f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} \\ f'(x) = \lim_{h \to 0} \frac{\sin(x + h) - \sin(x)}{h} \\ f'(x) = \lim_{h \to 0} \frac{\sin x \cos h + \cos x \sin h - \sin x}{h} \\ f'(x) = \lim_{h \to 0} \left ( \sin x \frac{(\cos h - 1)}{h} + \cos x \frac{\sin h}{h} \right ) \end{align}
• Now recall that $\lim_{h \to 0} \frac{(\cos h - 1)}{h} = 0$, and $\lim_{h \to 0} \frac{\sin h}{h} = 1$, so therefore, $f'(x) = 0 \cdot \sin x + 1 \cdot \cos x = \cos x$. $\blacksquare$
 Theorem 2: If $f(x) = \cos x$ then $\frac{d}{dx} f(x) = -\sin x$.
• Proof of Property (b): Let $f(x) = \cos x$. From the definition of a derivative it follows that:
(2)
\begin{align} f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} \\ f'(x) = \lim_{h \to 0} \frac{\cos(x + h) - \cos x}{h} \\ f'(x) = \lim_{h \to 0} \frac{\cos x \cos h - \sin x \sin h - \cos x}{h} \\ f'(x) = \lim_{h \to 0} \cos x \frac{(\cos h - 1)}{h} - \lim_{h \to 0} \sin x \frac{\sin h}{h} \\ f'(x) = 0 \cdot \cos x - 1 \cdot \sin x \\ f'(x) = -\sin x \quad \blacksquare \end{align}
 Theorem 3: If $f(x) = \tan x$ then $\frac{d}{dx} f(x) = \sec ^2 x$.

We will now prove theorem 3 at this point and time.

Example 1

Differentiate $f(x) = 3 \cos x + 2 \sin x$.

Applying the theorems above and we get that $f'(x) = -3 \sin x + 2 \cos x$.

Example 2

Differentiate $f(x) = 2\tan x - 2 \sin x$.

Applying the theorems above and we get that $f'(x) = 2 \sec ^2 x - 2 \cos x$.

Cyclic Pattern of Higher Order Derivatives of Sine and Cosine

Notice that the derivative of $\sin x$ is $\cos x$… the derivative of $\cos x$ is $-\sin x$, etc…. Hopefully you can see that there is a cycle here as:

(3)
\begin{align} \frac{d}{dx} \sin x = \cos x \end{align}
(4)
\begin{align} \frac{d}{dx} \cos x = -\sin x \end{align}
(5)
\begin{align} \frac{d}{dx} -\sin x = -\cos x \end{align}
(6)
\begin{align} \frac{d}{dx} -\cos x = \sin x \end{align}

We can thus make a general rule for higher order derivatives of $\sin x$ and $\cos x$. If $f(x) = \sin x$ or $f(x) = \cos x$ and $n = 4k + 1$, then for all integers $k ≥ 0$:

(7)
\begin{align} \frac{d^n}{dx^n} f(x) = f(x) \end{align}

Example 3

What function do you get if you differentiate $f(x) = \sin x$ one-hundred times?

Notice that every four differentiations, we will get back to $\sin x$. Therefore, if we differentiate $100$ times, we will still have $\sin x$.