Review your trigonometric function differentiation skills and use them to solve problems.

How do I differentiate trigonometric functions?

First, you should know the derivatives for the basic trigonometric functions:
start fraction, d, divided by, d, x, end fraction, sine, left parenthesis, x, right parenthesis, equals, cosine, left parenthesis, x, right parenthesis
start fraction, d, divided by, d, x, end fraction, cosine, left parenthesis, x, right parenthesis, equals, minus, sine, left parenthesis, x, right parenthesis
ddxtan(x)=sec2(x)=1cos2(x)\dfrac{d}{dx}\tan(x)=\sec^2(x)=\dfrac{1}{\cos^2(x)}
ddxcot(x)=csc2(x)=1sin2(x)\dfrac{d}{dx}\cot(x)=-\csc^2(x)=-\dfrac{1}{\sin^2(x)}
ddxsec(x)=sec(x)tan(x)=sin(x)cos2(x)\dfrac{d}{dx}\sec(x)=\sec(x)\tan(x)=\dfrac{\sin(x)}{\cos^2(x)}
ddxcsc(x)=csc(x)cot(x)=cos(x)sin2(x)\dfrac{d}{dx}\csc(x)=-\csc(x)\cot(x)=-\dfrac{\cos(x)}{\sin^2(x)}
You can actually use the derivatives of sine and cosine (along with the quotient rule) to obtain the derivatives of all the other functions.
Want to learn more about differentiating trigonometric functions? Check out this video about sine and cosine, this video about tangent and cotangent, and this video about secant and cosecant.

Practice set 1: sine and cosine

Problem 1.1
f, left parenthesis, x, right parenthesis, equals, 2, x, minus, 3, sine, left parenthesis, x, right parenthesis
f, prime, left parenthesis, x, right parenthesis, equals

Want to try more problems like this? Check out this exercise.

Practice set 2: tangent, cotangent, secant, and cosecant

Problem 2.1
Let f, left parenthesis, x, right parenthesis, equals, tangent, left parenthesis, x, right parenthesis.
Find f, prime, left parenthesis, start fraction, pi, divided by, 6, end fraction, right parenthesis.
Please choose from one of the following options.

Want to try more problems like this? Check out this exercise.
After you've mastered the derivatives of the basic trigonometric functions, you can differentiate trigonometric functions whose arguments are polynomials, like sec(3π2x)\sec\left(\dfrac{3\pi}{2}-x\right).

Practice set 3: general trigonometric functions

Problem 3.1
g, left parenthesis, x, right parenthesis, equals, sine, left parenthesis, 4, x, start superscript, 2, end superscript, plus, 3, x, right parenthesis
g, prime, left parenthesis, x, right parenthesis, equals, question mark
Please choose from one of the following options.

Want to try more problems like this? Check out this exercise.