### Differentiation: The derivative of standard functions

### The derivative of trigonometric functions

There are rules to determine the derivative of the trigonometric functions #\blue{\sin}(x), \green{\cos}(x)# and #\purple{\tan}(x)#. These rules only apply to trigonometric functions in radians.

The derivative of sine

\[\dfrac{\dd}{\dd x}\blue{\sin}(x)=\green{\cos}(x)\]

**Example**

\[\dfrac{\dd}{\dd x}(3\cdot\blue{\sin}(x))=3\cdot\green{\cos}(x)\]

The derivative of cosine

\[\dfrac{\dd}{\dd x}\green{\cos}(x)=-\blue{\sin}(x)\]

**Example**

\[\dfrac{\dd}{\dd x}(4\cdot\green{\cos}(x))=-4\cdot\blue{\sin}(x)\]

We can write the derivative of tangent in two ways.

The derivative of tangent

\[\begin{array}{rcl}\dfrac{\dd}{\dd x}\purple{\tan}(x)&=&\dfrac{1}{\green{\cos}(x) ^2}\\\dfrac{\dd}{\dd x}\purple{\tan}(x)&=&1+\purple{\tan} (x)^2\end{array}\]

**Example**

\[\begin{array}{rcl}\dfrac{\dd}{\dd x}(3\cdot\purple{\tan}(x))&=&\dfrac{3}{\green{\cos}(x)^2}\\\dfrac{\dd}{\dd x}(3\cdot\purple{\tan}(x))&=&3+3\cdot\purple{\tan}(x)^2\end{array}\]

To calculate the derivatives of trigonometric functions, we often need the *product rule* and *sum rule*. We can also use the *chain rule* for finding the derivatives of trigonometric functions.

\[\begin{array}{rcl}

\dfrac{\dd}{\dd x} \left(7\cdot \cos \left(x\right)-5\right) &=&\displaystyle \frac{\dd}{\dd x} \left(7 \cdot\cos(x)\right) +\frac{\dd}{\dd x} \left(-5\right) \\

&&\phantom{xxx}\blue{\text{sum rule}}\\

&=& \displaystyle7 \cdot \frac{\dd}{\dd x} \left(\cos(x)\right) +0 \\

&&\phantom{xxx}\blue{\text{constant rule and derivative of constant is }0}\\

&=& \displaystyle-7\cdot \sin \left(x\right) \\

&&\phantom{xxx}\blue{\text{the derivative of }\cos(x)\text{ is }-\sin \left(x\right)}\\

\end{array}\]

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