Derivatives of the six trig functions
The six trig functions and their derivatives
We’ve learned about the basic derivative rules, including chain rule, and now we want to shift our attention toward the derivatives of specific kinds of functions.
In this section we’ll be looking at the derivatives of trigonometric functions, and later on we’ll look at the derivatives of exponential and logarithmic functions.
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Trigonometric derivatives
There are six basic trig functions, and we should know the derivative of each one.
When we differentiate a trig function, we always have to apply chain rule. For instance, in ???y=\sin{x}???, the ???\sin??? and ???x??? are not multiplied together. Instead, the ???x??? is the argument of the sine function. What this means for us is that the argument is always the “inside function,” so when we differentiate, we need to always multiply by the derivative of the argument.
With these six basic trig functions, the argument is ???x???, and the derivative of ???x??? is ???1???, so applying chain rule and multiplying by ???1??? doesn’t change the value of the derivative.
But when the derivative of the argument is anything other than ???1???, then applying chain rule will of course have an actual effect on the value of the derivative. For instance, the derivative of ???y=\sin{(2x)}??? will be ???y'=2\cos{(2x)}???, and the derivative of ???y=\sec{(3x^2)}??? will be ???y'=6x\sec{(3x^2)}\tan{(3x^2)}???.
How the derivatives of sine and cosine are related, plus how to find the derivatives of all six trig functions
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Finding the derivatives of trig functions, and applying chain rule to the argument
Example
Find the derivative.
???y=4\cos{(2x)}???
The derivative of ???y=\cos{x}??? is ???y’=-\sin{x}???. The argument of ???2x??? has a derivative of ???2???, so the derivative will be
???y’=4(-\sin{(2x)})(2)???
???y’=-8\sin{(2x)}???
Let’s do another example with a little more going on.
Example
Find the derivative.
???y=8x^5-9\cot{(7x^4)}???
Dealing with one term at a time, and remembering to use chain rule to handle the derivative of ???-9\cot{(7x^4)}???, we get
???y'=8(5)x^{5-1}-9(-\csc^2{(7x^4)})(7(4)x^{4-1})???
???y'=40x^4-9(-\csc^2{(7x^4)})(28x^3)???
???y'=40x^4+252x^3\csc^2{(7x^4)}???
Let’s try one more.
the argument is always the “inside function,” so when we differentiate, we need to always multiply by the derivative of the argument.
Example
Find the derivative.
???y=\sec{(7x^3)}-7x^5\sin{x}+3\csc{(5x^7)}???
Let’s look at one term at a time. The derivative of ???\sec{(7x^3)}??? is
???(\sec{(7x^3)}\tan{(7x^3)})(7(3)x^{3-1})???
???(\sec{(7x^3)}\tan{(7x^3)})(21x^2)???
???21x^2\sec{(7x^3)}\tan{(7x^3)}???
To find the derivative of ???-7x^5\sin{x}???, we’ll need to use product rule. If ???f(x)=-7x^5??? and ???f'(x)=-35x^4???, and ???g(x)=\sin{x}??? and ???g'(x)=\cos{x}???, then we can plug directly into the product rule formula.
???f(x)g'(x)+f'(x)g(x)???
???(-7x^5)(\cos{x})+(-35x^4)(\sin{x})???
???-7x^5\cos{x}-35x^4\sin{x}???
The derivative of ???3\csc{(5x^7)}??? is
???(-3\csc{(5x^7)}\cot{(5x^7)})(5(7)x^{7-1})???
???(-3\csc{(5x^7)}\cot{(5x^7)})(35x^6)???
???-105x^6\csc{(5x^7)}\cot{(5x^7)}???
Putting these derivatives together, we get
???y'=21x^2\sec{(7x^3)}\tan{(7x^3)}-7x^5\cos{x}-35x^4\sin{x}-105x^6\csc{(5x^7)}\cot{(5x^7)}???
