Derivatives of Trigonometric Functions

1. The Foundation: Derivatives of Sine and Cosine from First Principles

To truly understand where the derivatives of trigonometric functions come from, we need to return to the fundamental definition of the derivative (the limit of a difference quotient). This process also reveals a critical requirement: for these differentiation rules to work, angles must be measured in radians.

The following video from MIT OpenCourseWare provides a complete, clear derivation. It's a classic piece of calculus instruction that builds a really solid foundation.

Let's summarise the key points from that deep dive.

The Crucial Limits

The entire derivation hinges on two limits, which were proven geometrically using the "Squeeze Theorem" (or "Sandwich Theorem"):

1. ${\lim }_{h\to 0}\frac{\sin (h)}{h}=1$
2. ${\lim }_{h\to 0}\frac{\cos (h)-1}{h}=0$

These limits are only valid when $h$ is in radians.

Deriving the Derivative of Sine

Using the definition of the derivative, $f^{\prime }(x)={\lim }_{h\to 0}\frac{f(x+h)-f(x)}{h}$:

1. Set up the limit:
   $\frac{d}{dx}\sin (x)={\lim }_{h\to 0}\frac{\sin (x+h)-\sin (x)}{h}$

2. Use the angle sum identity for sine: $\sin (A+B)=\sin (A)\cos (B)+\cos (A)\sin (B)$
   $={\lim }_{h\to 0}\frac{(\sin (x)\cos (h)+\cos (x)\sin (h))-\sin (x)}{h}$

3. Rearrange the terms to isolate the crucial limits:
   $={\lim }_{h\to 0}\left(\frac{\sin (x)\cos (h)-\sin (x)}{h}+\frac{\cos (x)\sin (h)}{h}\right)$
   $={\lim }_{h\to 0}\left(\sin (x)\left(\frac{\cos (h)-1}{h}\right)+\cos (x)\left(\frac{\sin (h)}{h}\right)\right)$

4. Evaluate the limits:
   $=\sin (x)\cdot (0)+\cos (x)\cdot (1)=\cos (x)$

Thus, we have formally proven that the derivative of $\sin (x)$ is $\cos (x)$.

A very similar process, using the angle sum identity for cosine, shows that the derivative of $\cos (x)$ is $-\sin (x)$.

2. Finding the Other Trigonometric Derivatives

Now for the good news: we don't need to go through that intense process for the other four functions. We can derive their derivatives simply by expressing them in terms of sine and cosine and applying the rules we already know.

Derivative of Tangent

Since $\tan (x)=\frac{\sin (x)}{\cos (x)}$, we can use the Quotient Rule, where $f(x)=\sin (x)$ and $g(x)=\cos (x)$.

Recall the Quotient Rule: ${\left(\frac{f}{g}\right)}^{\prime }=\frac{g\cdot f^{\prime }-f\cdot g^{\prime }}{g^2}$

So, the derivative of $\tan (x)$ is ${\sec }^2(x)$.

The remaining three derivatives can be found similarly:

• For $\cot (x)=\frac{\cos (x)}{\sin (x)}$: Use the Quotient Rule.
• For $\sec (x)=\frac{1}{\cos (x)}$: Use the Quotient Rule, or the Chain Rule on $(\cos (x){)}^{-1}$.
• For $\text{cosec}(x)=\frac{1}{\sin (x)}$: Use the Quotient Rule, or the Chain Rule on $(\sin (x){)}^{-1}$.

3. Summary of Rules and Practical Application

Let's gather all six rules into one place. This table is your reference for differentiating the basic trigonometric functions.

Of course, we rarely differentiate just $\sin (x)$. More often, we encounter functions like $\cos (5x)$ or $\tan (x^2)$. As we learned in the last lesson, this is where the Chain Rule is essential. The general form of the rule is:

If $y=\sin (f(x))$, then $\frac{dy}{dx}=\cos (f(x))\cdot f^{\prime }(x)$

This pattern—differentiate the outer trig function, keep the inner function the same, then multiply by the derivative of the inner function—applies to all six.

The following video provides a great summary of these rules in their general form and works through a variety of examples that combine them with the Chain, Product, and Quotient rules.

Let's work through one more example that brings everything together.

Find the derivative of $y=e^{2x}\tan (3x)$.

1. Identify the main structure: This is a product of two functions, $f(x)=e^{2x}$ and $g(x)=\tan (3x)$. So we start with the Product Rule.
   $y^{\prime }=f(x)g^{\prime }(x)+g(x)f^{\prime }(x)$
2. Find the derivatives of the parts:
   • $f^{\prime }(x)=\frac{d}{dx}(e^{2x})$. Using the Chain Rule, this is $e^{2x}\cdot 2=2e^{2x}$.
   • $g^{\prime }(x)=\frac{d}{dx}(\tan (3x))$. Using the Chain Rule on our new trig rule: the derivative of the outer function ($\tan$) is ${\sec }^2$, and the derivative of the inner function ($3x$) is $3$. So, $g^{\prime }(x)={\sec }^2(3x)\cdot 3=3{\sec }^2(3x)$.
3. Substitute back into the Product Rule:
   $y^{\prime }=(e^{2x})(3{\sec }^2(3x))+(\tan (3x))(2e^{2x})$
4. Simplify (optional but good practice): We can factor out the common term $e^{2x}$.
   $y^{\prime }=e^{2x}(3{\sec }^2(3x)+2\tan (3x))$

Conclusion

In this lesson, we have derived and learned to apply the derivatives for all six trigonometric functions. This is a major milestone in your calculus journey.

Key Takeaways:

• The derivatives of $\sin (x)$ and $\cos (x)$ are $\cos (x)$ and $-\sin (x)$, respectively. These foundational rules are derived from first principles.
• A critical prerequisite for all trigonometric differentiation is that angles must be measured in radians.
• The derivatives of $\tan (x)$, $\cot (x)$, $\sec (x)$, and $\text{cosec}(x)$ are ${\sec }^2(x)$, $-{\text{cosec}}^2(x)$, $\sec (x)\tan (x)$, and $-\text{cosec}(x)\cot (x)$. These can be derived from the sine and cosine rules using the Quotient and Chain Rules.
• For any trigonometric function with an inner function (e.g., $\sin (f(x))$), you must always apply the Chain Rule.

We have now covered differentiation for polynomials, powers, and trigonometric functions. In our next lesson, we will complete our toolkit by finding the derivatives for the remaining major function families: exponential and logarithmic functions.