Introduction to Trigonometric Differentiation
There are six basic trigonometric functions in geometry. The rate of change of these functions is known as trigonometric differentiation. This method follows the derivative rules and formulas to calculate the rate of change. Let’s understand how to apply trigonometric differentiation step-by-step and learn the difference between log and trig differentiation.
Understanding the Trigonometric Differentiation
A function that relates an angle of a right-angled triangle to its two sides is known as a trigonometric function. There are six basic trigonometric functions: cosine, tangent, cosecant, secant, and cotangent. In calculus, the derivative of trigonometric functions can be calculated using derivative rules.
By definition the trigonometric differentiation is defined as:
“The process of finding the derivative of a trigonometric function is called trigonometric differentiation.”
Trigonometric Differentiation Formulas
The derivative of a trigonometric function can be calculated by finding the rate of change of the sine and cosine functions. It is because knowing these two derivatives leads to the derivative of all other trig functions. The list of derivative formulas for trigonometric functions is as follows:
- $\frac{d}/dx(\sin x) = \cos x$
- $\frac{d}{dx}(\cos x) =\sin x$
- $\frac{d}{dx}(\sec x) = \sec x\tan x$
- $\frac{d}{dx}(\csc x) = -\csc x\cot x$
- $\frac{d}{dx}(\tan x) = \sec^2x$
- $\frac{d}{dx}(\cot x) = \csc²x$
The trigonometric differentiation formula can be modified to use it with other derivative rules. Let’s discuss the trig differentiation with the product rule, quotient, and power rules.
Trigonometric Differentiation and Product Rule
If a trig function is a product of two trig functions, the product rule with the trigonometric differentiation is used to calculate rate of change. For example, the derivative of secx tanx can be calculated as;
$\frac{d}{dx}(\sec x\tan x) = \sec x\frac{d}{dx}[\tan x] + \tan x\frac{d}{dx}[\sec x]$
Which is equal to,
$\frac{d}{dx}(\sec x\tan x) = \sec x(\sec^2x) + \tan x[\sec x\tan x]=\sec^3x+\tan^2xsec x$
Since $\sec^2x + \tan^2x = 1$ then,
$\frac{d}{dx}(\sec x\tan x)=\sec x(\sec^2x+\tan^2x)=\sec x$
Trigonometric Differentiation and Power Rule
Since the trigonometric differentiation is used to calculate derivatives of a trigonometric function. It can be used along with the power rule if the function contains a trig function with power n. The relation between power rule and trig differentiation for a function $f(x) = \cos^3x$ is expressed as;
$f’(x)=\frac{d}{dx}[\cos^3x]$
By using power rule formula,
$f’(x,y)=3\cos^{3-1}\frac{d}{dx}(\cos x)= -3\cos^2x\sin x$
Where, the cos^3x derivative with respect to x is $–3\cos^2x\sin x$.
Trigonometric Differentiation and Quotient Rule
If a trigonometric function is divided by another function, the trigonometric differentiation along with the quotient rule to find derivative. For a quotient of a function f(x) = tan x = sin x/cos x, the relation between trig derivative and quotient rule is,
$\frac{df}{dx}=\frac{\cos x\frac{d}{dx}[\sin x] – \sin x\frac{d}{dx}[\cos x]}{\cos^2x}$
And,
$\frac{df}{dx}=\frac{\cos^2x + \sin^2x}{\cos^2x} = \frac{1}{\cos^2x}=\sec^2x$
Hence the derivative of tan x is $\sec^2x$.
How do you do Trigonometric differentiation step by step?
The implementation of trigonometric derivatives is divided into a few steps. These steps assist us in calculating the derivative of a function having a trigonometric identity. These steps are:
- Write the expression of the function.
- Identify the trig function.
- Differentiate the function with respect to the variable involved.
- Use the trigonometric differentiation formula to calculate derivatives. For example, the derivative of sec x is tanx secx.
- Simplify if needed.
Applying logarithmic differentiation formula by using calculator
The derivative of a log function can also be calculated using the derivative calculator. The online tool follows the log differentiation formula to find the derivative. You can find it online by searching for a derivative calculator. For example, to calculate the derivative of ln x, the following steps are used by using this calculator.
- Write the expression of the function in the input box, such as ln x.
- Choose the variable to calculate the rate of change, which will be x in this example.
- Review the input so there will be no syntax error in the function.
- Now at the last step, click on the calculate button. By using this step, the derivative calculator will provide the derivative of ln x quickly and accurately, which will be 1/x
Comparison between Trigonometric and Logarithmic differentiation
The comparison between the logarithmic and trigonometric differentiation can be easily analysed using the following difference table.
Logarithmic Differentiation | Trigonometric Differentiation |
The logarithmic differentiation is used to calculate the derivative of a logarithmic function. | The trig differentiation is used to calculate derivative of a trigonometric function. |
The derivative of a logarithmic function ln x is defined as; $f’(x) = \frac{1}{x}\frac{d}{dx}(x)$ | There are different formulas to calculate derivative of trigonometric functions. |
The logarithmic differentiation can be used along with different derivative formulas. | The derivative of all trigonometric functions can be calculated by using product rule, quotient rule and power rule. |
Conclusion
The trigonometric functions are the functions that relate an angle with the right-angle triangle. Calculating the derivative of such functions is known as trig differentiation. In conclusion, the trigonometric differentiation of all trig functions can be calculated by using the derivatives of sine and cosine.