Trigonometric Functions

Derivatives of Basic Trigonometric Functions

The derivatives of the three fundamental trigonometric functions are essential in calculus and can be found in the log tables:

$$\frac{d}{dx} \left[ \cos(x) \right] = -\sin(x)$$ $$\frac{d}{dx} \left[ \sin(x) \right] = \cos(x)$$ $$\frac{d}{dx} \left[ \tan(x) \right] = \sec^2(x)$$

For more complex trigonometric functions, differentiation often requires the chain rule in addition to these basic derivative rules. This is particularly useful when differentiating functions of the form $\sin(g(x))$, $\cos(g(x))$, $\tan(g(x))$, where $g(x)$ is a function of $x$.

Example

Example

We can rewrite the expression as :

$$\left[ \sin(5x+2) \right]^2$$

Identify an inner and outer function :

$$v=\sin(5x+2)$$ $$u=v^2$$

Differentiate both functions :

$v'$

$u'$

Apply chain rule :

$$\begin{align*} u' \cdot v' &= 2v \cdot 5 \cos(5x+2) \\\\ &= 2(\sin(5x+2)) \cdot 5 \cos(5x+2) \\\\ &= 10\sin(5x+2)\cos(5x+2) \end{align*}$$

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