Trigonometric Derivatives — Practice Problems and Worked Solutions

The formulas you need here

16 worked problems

Every step is generated and checked by a computer algebra system.

1. 1.

   Differentiate the trigonometric function.

   $$f(x)=\sin \left(x\right)$$

   Answer: $\cos \left(x\right)$

   Step-by-step solution

   1. $$(\sin x{)}^{\prime }=\cos x,(\cos x{)}^{\prime }=-\sin x,(\tan x{)}^{\prime }={\sec }^{2}x$$
   2. $$f^{\prime }(x)=\cos \left(x\right)$$
2. 2.

   Differentiate the trigonometric function.

   $$f(x)=\cos \left(x\right)$$

   Answer: $-\sin \left(x\right)$

   Step-by-step solution

   1. $$(\sin x{)}^{\prime }=\cos x,(\cos x{)}^{\prime }=-\sin x,(\tan x{)}^{\prime }={\sec }^{2}x$$
   2. $$f^{\prime }(x)=-\sin \left(x\right)$$
3. 3.

   Differentiate the trigonometric function.

   $$f(x)=\tan \left(x\right)$$

   Answer: $\frac{1}{{\cos }^2\left(x\right)}$

   Step-by-step solution

   1. $$(\sin x{)}^{\prime }=\cos x,(\cos x{)}^{\prime }=-\sin x,(\tan x{)}^{\prime }={\sec }^{2}x$$
   2. $$f^{\prime }(x)=\frac{1}{{\cos }^2\left(x\right)}$$
4. 4.

   Differentiate the trigonometric function.

   $$f(x)=3\sin \left(x\right)-2\cos \left(x\right)$$

   Answer: $2\sin \left(x\right)+3\cos \left(x\right)$

   Step-by-step solution

   1. $$(\sin x{)}^{\prime }=\cos x,(\cos x{)}^{\prime }=-\sin x,(\tan x{)}^{\prime }={\sec }^{2}x$$
   2. $$f^{\prime }(x)=2\sin \left(x\right)+3\cos \left(x\right)$$
5. 5.

   Differentiate the trigonometric function.

   $$f(x)=x+\sin \left(x\right)$$

   Answer: $\cos \left(x\right)+1$

   Step-by-step solution

   1. $$(\sin x{)}^{\prime }=\cos x,(\cos x{)}^{\prime }=-\sin x,(\tan x{)}^{\prime }={\sec }^{2}x$$
   2. $$f^{\prime }(x)=\cos \left(x\right)+1$$
6. 6.

   Differentiate the trigonometric function.

   $$f(x)=\cos \left(x\right)+2\tan \left(x\right)$$

   Answer: $-\sin \left(x\right)+\frac{2}{{\cos }^2\left(x\right)}$

   Step-by-step solution

   1. $$(\sin x{)}^{\prime }=\cos x,(\cos x{)}^{\prime }=-\sin x,(\tan x{)}^{\prime }={\sec }^{2}x$$
   2. $$f^{\prime }(x)=-\sin \left(x\right)+\frac{2}{{\cos }^2\left(x\right)}$$
7. 7.

   Differentiate the trigonometric function.

   $$f(x)=5\sin \left(x\right)$$

   Answer: $5\cos \left(x\right)$

   Step-by-step solution

   1. $$(\sin x{)}^{\prime }=\cos x,(\cos x{)}^{\prime }=-\sin x,(\tan x{)}^{\prime }={\sec }^{2}x$$
   2. $$f^{\prime }(x)=5\cos \left(x\right)$$
8. 8.

   Differentiate the trigonometric function.

   $$f(x)=4\cos \left(x\right)$$

   Answer: $-4\sin \left(x\right)$

   Step-by-step solution

   1. $$(\sin x{)}^{\prime }=\cos x,(\cos x{)}^{\prime }=-\sin x,(\tan x{)}^{\prime }={\sec }^{2}x$$
   2. $$f^{\prime }(x)=-4\sin \left(x\right)$$
9. 9.

   Differentiate the trigonometric function.

   $$f(x)=\sin \left(x\right)+\cos \left(x\right)$$

   Answer: $-\sin \left(x\right)+\cos \left(x\right)$

   Step-by-step solution

   1. $$(\sin x{)}^{\prime }=\cos x,(\cos x{)}^{\prime }=-\sin x,(\tan x{)}^{\prime }={\sec }^{2}x$$
   2. $$f^{\prime }(x)=-\sin \left(x\right)+\cos \left(x\right)$$
10. 10.

    Differentiate the trigonometric function.

    $$f(x)=3x+2\sin \left(x\right)$$

    Answer: $2\cos \left(x\right)+3$

    Step-by-step solution

    1. $$(\sin x{)}^{\prime }=\cos x,(\cos x{)}^{\prime }=-\sin x,(\tan x{)}^{\prime }={\sec }^{2}x$$
    2. $$f^{\prime }(x)=2\cos \left(x\right)+3$$
11. 11.

    Differentiate the trigonometric function.

    $$f(x)=x^2+\cos \left(x\right)$$

    Answer: $2x-\sin \left(x\right)$

    Step-by-step solution

    1. $$(\sin x{)}^{\prime }=\cos x,(\cos x{)}^{\prime }=-\sin x,(\tan x{)}^{\prime }={\sec }^{2}x$$
    2. $$f^{\prime }(x)=2x-\sin \left(x\right)$$
12. 12.

    Differentiate the trigonometric function.

    $$f(x)=3\tan \left(x\right)$$

    Answer: $\frac{3}{{\cos }^2\left(x\right)}$

    Step-by-step solution

    1. $$(\sin x{)}^{\prime }=\cos x,(\cos x{)}^{\prime }=-\sin x,(\tan x{)}^{\prime }={\sec }^{2}x$$
    2. $$f^{\prime }(x)=\frac{3}{{\cos }^2\left(x\right)}$$
13. 13.

    Differentiate the trigonometric function.

    $$f(x)=\sin \left(x\right)-4\cos \left(x\right)$$

    Answer: $4\sin \left(x\right)+\cos \left(x\right)$

    Step-by-step solution

    1. $$(\sin x{)}^{\prime }=\cos x,(\cos x{)}^{\prime }=-\sin x,(\tan x{)}^{\prime }={\sec }^{2}x$$
    2. $$f^{\prime }(x)=4\sin \left(x\right)+\cos \left(x\right)$$
14. 14.

    Differentiate the trigonometric function.

    $$f(x)=x\sin \left(x\right)$$

    Answer: $x\cos \left(x\right)+\sin \left(x\right)$

    Step-by-step solution

    1. $$(\sin x{)}^{\prime }=\cos x,(\cos x{)}^{\prime }=-\sin x,(\tan x{)}^{\prime }={\sec }^{2}x$$
    2. $$f^{\prime }(x)=x\cos \left(x\right)+\sin \left(x\right)$$
15. 15.

    Differentiate the trigonometric function.

    $$f(x)=\sin \left(x\right)\cos \left(x\right)$$

    Answer: $\cos \left(2x\right)$

    Step-by-step solution

    1. $$(\sin x{)}^{\prime }=\cos x,(\cos x{)}^{\prime }=-\sin x,(\tan x{)}^{\prime }={\sec }^{2}x$$
    2. $$f^{\prime }(x)=\cos \left(2x\right)$$
16. 16.

    Differentiate the trigonometric function.

    $$f(x)=\sin \left(x\right)+\tan \left(x\right)$$

    Answer: $\cos \left(x\right)+\frac{1}{{\cos }^2\left(x\right)}$

    Step-by-step solution

    1. $$(\sin x{)}^{\prime }=\cos x,(\cos x{)}^{\prime }=-\sin x,(\tan x{)}^{\prime }={\sec }^{2}x$$
    2. $$f^{\prime }(x)=\cos \left(x\right)+\frac{1}{{\cos }^2\left(x\right)}$$

More from Calculus I

• Evaluating Limits
• One-Sided Limits
• Limits at Infinity
• Continuity
• Derivative from First Principles
• Power Rule
• Exponential & Log Derivatives
• Product Rule
• Quotient Rule
• Chain Rule
• Implicit Differentiation
• Related Rates
• Curve Analysis
• Optimization
• Antiderivatives
• Definite Integrals
• Fundamental Theorem of Calculus
• u-Substitution
• Area Between Curves
• Volumes of Revolution