Why the derivative of sin(ax) is a·cos(ax)
With u = ax and du/dx = a, the chain rule gives d/dx[sin(u)] = cos(u) · a = a·cos(ax).
Examples: d/dx[sin(2x)] = 2·cos(2x), d/dx[sin(x/3)] = ⅓·cos(x/3).
∫
Calculus
University
Periodic Circular & Harmonic Ratio
Derivative of sin(ax)
Compute the derivative of sin(ax) at a specific x value (x in radians).
Standard Formula
d/dx[sin(ax)] = a·cos(ax)
Interactive Sandbox
Live Reactive
Adjust values in real-time and observe mathematical cause-and-effect
a
0.120
x
120
Live EquationFormula Proof
d/dx[sin(ax)] = 1·cos(ax) = 1
Tangent Line dy/dx
Calculated Solution
1
Bounded Oscillatory Function: Sine and cosine oscillate smoothly between -1 and +1 with period 360° (2π rad).
Variable Mechanics
Roles, constraints & impacts
2 Params
a
Coefficient
Formula Parameter
ImpactDirect contribution
ConstraintReal numbers
x
Value of x (radians)
Input Angle / Ratio
ImpactEvaluates trigonometric projection
ConstraintReal number
At a Glance
Gotchas, units & real-world use
Essential
Formula Nature & Mechanics
Bounded Oscillatory Function: Sine and cosine oscillate smoothly between -1 and +1 with period 360° (2π rad).
Common Mistakes & Traps
- Using Degree mode when Radians are expected (or vice versa).
- Tangent asymptotes at 90° and 270° (division by cos θ = 0 causes undefined infinity).
- Forgetting inverse trigonometric domain limits (e.g., arcsin input must be between -1 and 1).
Dimensional Analysis & Units
Ratio of side lengths → Dimensionless Scalar [-1, +1]
Real-World & Industry Application
Audio sound wave synthesis, AC alternating current electrical waveforms, robotic arm joint kinematics, and astronomical navigation.
Worked Example: Derivative of sin(ax)
Given:
- a = 1
- x = 0
- d/dx[sin(x)] = cos(x)
- At x=0: cos(0) = 1
Answer: 1
Frequently Asked Questions
What is the Derivative of sin(ax) formula?
d/dx[sin(ax)] = a·cos(ax). Compute the derivative of sin(ax) at a specific x value (x in radians).
How do I calculate Derivative of sin(ax)?
Enter Coefficient (a), Value of x (radians) (x) into the calculator. It applies d/dx[sin(ax)] = a·cos(ax) and shows every step of the working.
What is the derivative of sin(ax)?
a·cos(ax), by the chain rule.
Related Calculus Calculators
Power Rule for Derivativesd/dx(xⁿ) = n × x^(n-1)Definite Integral (Power Rule)∫[a to b] x^n dx = [x^(n+1)/(n+1)]|[a to b]Partial Derivative∂f/∂x = lim[h→0] (f(x+h,y) - f(x,y))/hChain Rule (Derivative)d/dx[f(g(x))] = f'(g(x)) · g'(x)Product Rule (Derivative)d/dx[f·g] = f'·g + f·g'Quotient Rule (Derivative)d/dx[f/g] = (f'·g − f·g') / g²
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