π Table of Contents
What is the derivative of sinx? The quick answer
If you have ever asked yourself “what is the derivative of sinx?,” the answer is straightforward: it is cosx. In calculus notation,
$$ \frac{d}{dx} \sin x = \cos x $$
This means that at any point \(x\) (measured in radians), the instantaneous rate of change of the sine function equals the cosine of that same angle. For example, at \(x=0\), sinx has a slope of \( \cos0 = 1 \). At \(x=\pi/2\), the slope is \( \cos(\pi/2) = 0\), which matches the flat top of the sine curve. Knowing what is the derivative of sinx is essential for solving problems in physics, engineering, and every branch of calculus that involves periodic motion.
Less common but equally important: if the argument is a function of x (like sin(3x)), you apply the chain rule. The core rule that the derivative of sinx is cosx never changes β it just becomes the “outer” derivative.
π Key Takeaways
- The derivative of sinx is cosx β memorise this as a core rule.
- This result comes from the limit definition of derivative and trigonometric identities.
- Always use radians. In degrees, the derivative would involve an extra factor.
- Apply the chain rule when the argument is more complex, e.g., \( \sin(2x) \) differentiates to \( 2\cos(2x) \).
- The geometric interpretation: the slope of the sine curve at any point equals the cosine value at that point.
Understanding why the derivative of sinx is cosx
To truly grasp what is the derivative of sinx, we need to look at the limit definition. Recall that the derivative of a function \(f\) at \(x\) is
$$ f'(x) = \lim_{h \to 0} \frac{f(x+h) – f(x)}{h} $$
Apply this to \( f(x)=\sin x \):
$$ \frac{d}{dx} \sin x = \lim_{h\to 0} \frac{\sin(x+h) – \sin x}{h} $$
Using the angle-sum identity, \(\sin(x+h) = \sin x \cos h + \cos x \sin h\). Substituting gives
$$ \lim_{h\to 0} \frac{\sin x \cos h + \cos x \sin h – \sin x}{h} = \sin x \cdot \lim_{h\to 0} \frac{\cos h – 1}{h} + \cos x \cdot \lim_{h\to 0} \frac{\sin h}{h} $$
The two limits are well-known: \(\lim_{h\to 0} \frac{\sin h}{h} = 1\) and \(\lim_{h\to 0} \frac{\cos h – 1}{h} = 0\). The first limit can be proven using the squeeze theorem and a geometric argument with the unit circle: for small h (in radians), the area of a sector bounds the ratio. The second limit follows from the first using the identity \(1-\cos h = 2\sin^2(h/2)\). Therefore,
$$ \frac{d}{dx} \sin x = (\sin x \cdot 0) + (\cos x \cdot 1) = \cos x $$
This confirms exactly what is the derivative of sinx β it is cosx. The key step uses the squeeze theorem to prove \(\lim_{h\to 0} \sin h / h = 1\), which only holds when \(h\) is in radians. If you attempt the same limit with degrees, you’d get \(\pi/180\) instead of 1, altering the derivative.
This derivation is not just a formality. Understanding it helps you see why the derivative of cosx is -sinx (a similar limit proof ends with a minus sign) and prepares you for more advanced topics like the derivative of inverse trig functions.
What is the derivative of sinx? A visual intuition
Imagine the graph of \(y=\sin x\). Its slope changes continuously: steep at the origin, flat at the peaks and troughs. The cosine function exactly describes that slope. At \(x=0\), \(\cos 0=1\) (rising at 45Β°); at \(x=\pi\), \(\cos\pi = -1\) (falling). This geometric harmony is why what is the derivative of sinx is such a neat rule. You can even approximate derivatives by drawing tangent lines: the slope of the tangent to sinx at any x equals the height of the cosine curve at that x.
Using the unit circle to understand the derivative of sinx
On the unit circle, \(\sin\theta\) is the y-coordinate. As \(\theta\) increases, the y-coordinate changes at a rate proportional to the x-coordinate β which is exactly \(\cos\theta\). So what is the derivative of sinx? It’s the projection of motion onto the vertical axis, and it’s \(\cos x\). This insight extends to parametric curves: the derivative of the vertical component of uniform circular motion is the horizontal component. For a deeper intuition, check out this Khan Academy explanation (external resource with additional examples).
Common mistakes when finding the derivative of sinx
Here are the top three errors students make when asked “what is the derivative of sinx?”
- Forgetting radians: If you use degrees, the derivative gains an extra factor of \(\pi/180\). Always set your calculator to radians when differentiating.
- Confusing with the derivative of cosine: The derivative of cosx is -sinx, not -cosx. Many students remember the sign wrong.
- Misapplying the product rule: For a product like \(x^2 \sin x\), you must use the product rule: derivative = \(2x \sin x + x^2 \cos x\). The derivative of sinx alone is still cosx, but the whole expression requires both parts.
Another subtle mistake: assuming the derivative of sinx is cosx even when x is not the variable. For example, the derivative of sin(y) with respect to t requires the chain rule: cos(y)Β·dy/dt. Always check what variable you are differentiating with respect to.
What is the derivative of sinx? 5 Worked examples with real numbers
To solidify what is the derivative of sinx, we work through five examples of increasing complexity. Each example applies the core rule and shows how it behaves in different contexts.
π§ͺ Worked example 1: Slope at a specific point
Solution: The derivative of sinx is cosx. Evaluate at \(x=\pi/3\):
\(\cos(\pi/3) = 1/2\). So the slope at that point is 0.5. This means the sine curve is rising at exactly half its maximum slope (which is 1 at the origin).
π§ͺ Worked example 2: Using the chain rule
Solution: Let \(u = 2x+1\). Then \(dy/dx = \cos(2x+1) \cdot 2 = 2\cos(2x+1)\). The derivative of sinx inside the chain rule still behaves the same: the outer derivative is cos(u). If the argument were \(\sin(x^2)\), the derivative would be \(2x \cos(x^2)\). The pattern is always: derivative of outer (cos) times derivative of inner.
π§ͺ Worked example 3: Real-world application β simple harmonic motion
Solution: Velocity \(v(t)=ds/dt = 5\cdot 3\cos(3t) = 15\cos(3t)\). At \(t=2\), \(v=15\cos(6) \approx 15(0.960) = 14.4\) cm/s. The derivative of sinx gives the instantaneous velocity. Without this rule, we could only approximate velocity numerically.
π§ͺ Worked example 4: Combining with product rule
Solution: Use product rule: \(y’ = (1)\sin x + x(\cos x) = \sin x + x\cos x\). The derivative of sinx appears as the first term in the sum. If we incorrectly thought the derivative of sinx were something else, the product rule would be wrong.
π§ͺ Worked example 5: Second derivative of sinx
Solution: First derivative: \(f'(x) = \cos x\). Second derivative: \(f”(x) = -\sin x\) (since derivative of cosx is -sinx). So the second derivative of sinx is -sinx, which explains the oscillatory nature of acceleration in harmonic motion. Knowing what is the derivative of sinx allows us to build derivatives of any order.
What is the derivative of sinx? Pros and cons of memorising vs deriving
β Pros of memorising
- Fast and efficient in exams
- Lets you focus on more complex differentiation problems
- Builds intuition for derivative patterns
β Cons of only memorising
- May forget under pressure if you don’t understand the proof
- Harder to extend to related functions like \(\sin(kx)\)
- Less ability to spot errors in your work
I recommend understanding the proof once, then memorising the result. That way, when someone asks you “what is the derivative of sinx?” you can answer confidently and know why it’s true. The extra effort to understand the limit proof pays dividends when you later encounter the derivative of inverse sine or the integration of sine.
At-a-glance: Derivative of sinx compared to other trig derivatives
| Function | Derivative | Why it matters |
|---|---|---|
| sinx | cosx | Foundation for all periodic motion derivatives |
| cosx | -sinx | Note the negative sign |
| tanx | secΒ²x | Derived from quotient rule (sinx/cosx) |
| cscx | -cscx cotx | Chain rule with 1/sinx |
Chain rule connection: derivative of sin(u)
Once you know what is the derivative of sinx, differentiating \(\sin(u)\) where u is a function of x is straightforward via the chain rule:
$$ \frac{d}{dx} \sin(u) = \cos(u) \cdot \frac{du}{dx} $$
For example, \(\frac{d}{dx} \sin(5x) = 5\cos(5x)\). This rule is used everywhere from wave equations to economics models (e.g., seasonal cycles). The key is to always take the derivative of the outer sine (which is cosine) and multiply by the derivative of the inner function. If you forget the chain rule, your answer for something like \(\sin(x^3)\) would be missing the factor \(3x^2\).
Practical applications of the derivative of sinx
Understanding what is the derivative of sinx opens the door to real-world problem solving:
- Physics: Simple harmonic motion (mass on a spring, pendulum) uses \(s(t)=A\sin(\omega t)\). Velocity and acceleration require differentiating sine.
- Engineering: AC circuit analysis involves voltage \(V(t)=V_0\sin(\omega t)\). The derivative gives the current through a capacitor.
- Signal processing: The derivative of a sinusoidal signal gives its rate of change, used in edge detection and audio processing.
- Economics: Seasonal trends can sometimes be modeled by sine functions; derivatives help find turning points (peaks and troughs).
For instance, if the altitude of a sound wave is modeled by \(p(t)=0.01\sin(440t)\), the rate of pressure change is \(p'(t)=4.4\cos(440t)\) pascals per second. Without the derivative rule, you would need to approximate this numerically at every instant.
Frequently asked questions about the derivative of sinx
Still wondering about what is the derivative of sinx? These answers cover common lingering questions.
β What is the derivative of sinx?
The derivative of sinx is cosx. This holds for all real x when measured in radians.
β Why is the derivative of sinx equal to cosx?
It follows from the limit definition of derivative and the trigonometric identities \(\lim_{h\to 0} \frac{\sin h}{h} = 1\) and \(\lim_{h\to 0} \frac{\cos h – 1}{h} = 0\).
β Does the derivative of sinx change if x is in degrees?
Yes. In degrees, the derivative is (\(\pi/180)\cos x\). Always use radians for the standard rule.
β How do I differentiate sin(2x)?
Use the chain rule: derivative = \(2\cos(2x)\). The outer derivative of sin(u) is cos(u), multiplied by the derivative of u=2x.
β What is the derivative of sinx at x=0?
At x=0, the derivative is cos(0)=1. The sine curve rises with slope 1 at the origin.
For further reading, the Wolfram MathWorld article on sine (external DoFollow link) provides a comprehensive overview of the sine function and its properties, including the derivative.
Now you have a complete picture of what is the derivative of sinx. Practice the rule, apply the chain rule when needed, and you’ll be ready to handle any differentiation problem involving sine. If you’d like to go deeper, try deriving the derivative of cosx using the same limit method.