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1. Derivatives of the Sine, Cosine and Tangent Functions

by M. Bourne

Making use of an important result:

it can be shown from first principles that:

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:

Here is a demonstration of what the derivative of a transcendental curve really means, using LiveMath.

LIVEMath

 

Example 1:

Find the derivative of y = sin(x² + 3).

Here is the way LiveMath performs this derivative.

LIVEMath

Normal answer:

Answer

---

IMPORTANT:

cos x² + 3 ≠ cos(x² + 3).

Can you see why they must be different?

Example 2:

Find the derivative of y = cos 3x⁴.

Answer

 

Example 3:

Find the derivative of y = cos³2x

Answer

 

Example 4:

Find the derivative of y = 3 sin 4x + 5 cos 2x³.

Answer

 

Exercises:

1. Find the derivative of y = 4 cos (6x² + 5).

Answer

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

Answer

3. Find the derivative of y = (x − cos²x)⁴.

Here it is in LiveMath.

LIVEMath

Normal answer:

Answer

4. Find the derivative of:

Answer

5. Find the derivative of 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

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

Answer

9. Show that y = cos³x tan x satisfies

Answer

10. Find the derivative of y = x tan x

Answer

Need a break? Play a math game. Well, they all involve math... No, really!

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

• Comments, Questions?

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