Chapter Contents

• Differentiation of Transcendental Functions
• 1. Derivatives of Sin, Cos and Tan Functions
• 2. Derivatives of Csc, Sec and Cot Functions
• 3. Derivatives of Inverse Trigonometric Functions
• 4. Applications: Derivatives of Trigonometric Functions
• 5. Derivative of the Logarithmic Function
• 6. Derivative of the Exponential Function
• 7. Applications: Derivatives of Logarithmic and Exponential Functions
• Differentiation of Transcendental Functions Problem Solver

• Comments, Questions?

Summer holidays

Recommendation

Easy to understand calculus lessons on DVD. Try before you commit. More info:
MathTutorDVD.com

Online Algebra Solver

Solve your algebra problem step by step!

Online Algebra Solver »

Share

Share this page.

1. Derivatives of the Sine, Cosine and Tangent Functions

by M. Bourne

It can be shown from first principles that:

`(d(sin\ x))/(dx)=cos\ x`

`(d(cos\ x))/dx=-sin\ x`

`(d(tan\ x))/(dx)=sec^2x`

In words, we would say:

The derivative of sin x is cos x,
The derivative of cos x is −sin x (note the negative sign!) and
The derivative of tan x is sec²x.

Now, if u = f(x) is a function of x, then by using the chain rule, we have:

`(d(sin\ u))/(dx)=cos\ u(du)/(dx)`

`(d(cos\ u))/dx=-sin\ u(du)/(dx)`

`(d(tan\ u))/(dx)=sec^2u(du)/(dx)`

Example 1

Differentiate `y = sin(x^2 + 3)`.

Answer

Example 2

Find the derivative of `y = cos\ 3x^4`.

Answer

Example 3

Differentiate `y = cos^3 2x`

Answer

Example 4

Find the derivative of `y = 3\ sin\ 4x + 5\ cos\ 2x^3`.

Answer

Exercises

1. Differentiate y = 4 cos (6x² + 5).

Answer

2. Find the derivative of y = 3 sin³ (2x⁴ + 1).

Answer

3. Differentiate y = (x − cos²x)⁴.

Answer

4. Find the derivative of:

`y=(2x+3)/(sin\ 4x)`

Answer

5. Differentiate y = 2x sin x + 2 cos x − x²cos x.

Answer

6. Find the derivative of the implicit function

x cos 2y + sin x cos y = 1.

Answer

7. Find the slope of the line tangent to the curve of

`y=(2\ sin\ 3x)/x`

where `x = 0.15`

Answer

8. The current (in amperes) in an amplifier circuit, as a function of the time t (in seconds) is given by

`i = 0.10\ cos (120πt + π/6)`.

Find the expression for the voltage across a 2.0 mH inductor in the circuit, given that

`V_L=L(di)/(dt)`

Answer

9. Show that y = cos³x tan x satisfies

`cos\ x(dy)/(dx)+3y\ sin\ x-cos^2x=0`

Answer

10. Find the derivative of y = x tan x

Answer

See also: Derivative of square root of sine x by first principles.