How to Find the Derivative of Sin X Using First Principles?
The concept of Derivative of Sin X is essential in mathematics and helps in solving real-world and exam-level problems efficiently. Understanding how to differentiate trigonometric functions like sin x makes calculus much easier for students, especially when preparing for board exams or competitive tests.
Understanding Derivative of Sin X
A derivative of sin x refers to the process of finding how the sine function changes with respect to x. In calculus, trigonometric functions like sine, cosine, and tangent play a major role. The derivative of sin x is commonly used in calculus, trigonometric differentiation, and rate of change problems. It forms a base for topics like limits and derivatives, chain rule, and graphical analysis in mathematics.
Formula Used in Derivative of Sin X
The standard formula is: \( \frac{d}{dx}[\sin x] = \cos x \)
Here’s a helpful table to understand derivative results for sine and related functions:
Derivative of Sin X Table
| Function | Derivative | Method Used |
|---|---|---|
| sin x | cos x | Basic Rule |
| sin(ax) | a cos(ax) | Chain Rule |
| sin x² | 2x cos(x²) | Chain Rule |
| sin(x) cos(x) | cos²x - sin²x | Product Rule |
This table shows the standard derivatives involving the sine function that often appear in board exams and competitive entrance tests.
Derivative of Sin X by First Principle (Limit Definition)
- Start with the first principle:
\( f'(x) = \lim_{h \to 0} \frac{\sin(x+h) - \sin(x)}{h} \) - Use the trigonometric identity:
\( \sin(x+h) = \sin x \cos h + \cos x \sin h \) - Substitute and rearrange:
\( = \lim_{h\to0} \frac{[\sin x (\cos h - 1) + \cos x \sin h]}{h} \) - Simplify and split the limit:
\( = \sin x \lim_{h\to0} \frac{\cos h - 1}{h} + \cos x \lim_{h\to0} \frac{\sin h}{h} \) - Apply known limits (limits and derivatives):
\( \lim_{h\to0} \frac{\sin h}{h} = 1 \), and \( \lim_{h\to0} \frac{\cos h-1}{h}\) = 0 - Final result:
\( f'(x) = \cos x \)
Graphical Explanation
If you plot the graph of \( y = \sin x \), its slope at each point is given by \( \cos x \). The graph of \( \cos x \) itself is the derivative—showing how the sine curve increases or decreases at every value. Use the Derivative Plotter to visualize this instantly.
Worked Example – Solving a Problem
- Find the derivative of sin(2x) with respect to x.
Let \( f(x) = \sin(2x) \)
Using the chain rule: \( \frac{d}{dx} [\sin(2x)] = 2\cos(2x) \) - Differentiate sin(x)·cos(x) with respect to x:
Product rule: \( d/dx [\sin x \cos x] = \cos x \cos x + \sin x (-\sin x) = \cos^2 x - \sin^2 x \)
Practice Problems
- Find the derivative of sin x⁴ with respect to x.
- Differentiated sin(x/2).
- Use the first principle to prove the derivative of sin x is cos x.
- What is the derivative of sin(xy) with respect to x?
- Sketch the graphs of y = sin x and y = cos x on the same axes.
Common Mistakes to Avoid
- Confusing derivative of sin x with the derivative of sin²x.
- Forgetting to use the chain rule for expressions like sin(x²), sin(ax), or sin(x/2).
- Mixing up radians and degrees—derivatives assume x is in radians.
- Ignoring negative signs or missing product/chain rules for composite functions.
Real-World Applications
The derivative of sin x appears in physics (wave motion, oscillations), engineering (signal processing), and even in economics (cyclical trends). In school and competitive maths, it is used in almost every question involving rate of change and motion. Vedantu helps students see how these mathematical rules support many real-life applications.
We explored the idea of derivative of sin x, how to apply it, solve related problems, and understand its real-life relevance. Practice more with Vedantu to build confidence in these concepts.
Further Learning & Related Resources
- Differentiation Formula – Complete list of basic and trigonometric differentiation rules.
- Chain Rule – Crucial for functions like sin(x²), sin(ax), and all composite derivatives.
- Product Rule – Used for derivatives like sin x cos x and more.
- Trigonometric Functions – Key formulae for sin, cos, tan and their derivatives and integrals.
- Limits and Derivatives – Builds your fundamentals for proving derivatives from scratch.
- Continuity and Differentiability – Ensures understanding of where the derivative of sin x is defined and why it’s continuous everywhere.
- Taylor Series – Learn how multiple derivatives are used to approximate sin x, cos x as polynomials.
- Derivative Plotter – Instantly visualize graphs of sin x and its derivatives online.
FAQs on Derivative of Sin X – Formula, Proof, Examples, and Applications
1. What is the derivative of sin x?
The derivative of sin x with respect to x is cos x. In notation, d/dx [sin x] = cos x. This fundamental result helps in solving calculus problems involving trigonometric functions.
2. How do you find the derivative of sin x using the first principle method?
Using the first principle of derivatives, the derivative of sin x is found by evaluating the limit: f'(x) = lim(h→0) [sin(x+h) - sin x]/h. By applying trigonometric identities and limit properties, this limit evaluates to cos x.
3. What is the derivative of sin (x) squared or sin2x?
The derivative of sin2x requires the chain rule. If f(x) = sin2x = (sin x)2, then f'(x) = 2 sin x · cos x.
4. How do you differentiate sin x/2?
To differentiate sin (x/2), apply the chain rule: d/dx [sin (x/2)] = cos (x/2) · (1/2) = (1/2) cos (x/2).
5. What is the derivative of sin x cos x?
Use the product rule to differentiate sin x cos x:
d/dx [sin x cos x] = cos x · cos x + sin x · (–sin x) = cos2x – sin2x, which simplifies to cos 2x by trigonometric identities.
6. How is the derivative of sin x represented graphically?
The graph of sin x is a smooth wave-like curve, and its derivative cos x also forms a similar wave, but shifted. The derivative graph shows the rate of change of sin x at each point.
7. Why is the derivative of sin x not equal to sin x itself?
The derivative measures the rate of change of a function. Since sin x is periodic and oscillates, its rate of change is given by cos x, which reflects how fast sin x increases or decreases, not the function value itself.
8. Why do students often confuse the derivative of sin x with that of sin2x?
Confusion arises because sin x and sin2x look similar but have different derivatives. The latter requires applying the chain rule, while the former has a straightforward derivative of cos x. Recognizing the function inside the sine is key.
9. What happens if you forget to apply the chain rule when differentiating sin x2?
Omitting the chain rule leads to incorrect derivatives. For example, the derivative of sin x2 is not cos x2, but 2x · cos x2. Forgetting the inner derivative factor causes calculation errors.
10. How does the derivative of sin x change if the angle is measured in degrees instead of radians?
If x is in degrees, the derivative of sin x becomes cos x multiplied by π/180 because the radian conversion factor must be applied. Thus, d/dx [sin x (degrees)] = (π/180) cos x.
11. Why does the proof of the derivative of sin x using first principles require the limit limθ→0 (sin θ)/θ = 1?
The limit limθ→0 (sin θ)/θ = 1 is crucial in the first principle derivation because it allows simplification of terms involving sine functions with small angles, ensuring the derivative of sin x correctly evaluates to cos x.
12. What are common mistakes when differentiating sin xy with respect to x?
Common errors include ignoring that sin xy is a product of variables, leading to missed application of the product rule and chain rule. The correct derivative is cos (xy) · (y + x · dy/dx) if y is a function of x.
