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Calculus

University

Periodic Circular & Harmonic Ratio

Derivative of cos(ax)

Compute the derivative of cos(ax) at a specific x value (x in radians).

Standard Formula

d/dx[cos(ax)] = −a·sin(ax)

Interactive Sandbox

Live Reactive

Adjust values in real-time and observe mathematical cause-and-effect

Coefficient

0.120

Value of x (radians)(Real number)

120

Live EquationFormula Proof

d/dx[cos(ax)] = −1·sin(ax) = 0

Tangent Line dy/dx

Calculated Solution

Bounded Oscillatory Function: Sine and cosine oscillate smoothly between -1 and +1 with period 360° (2π rad).

Variable Mechanics

Roles, constraints & impacts

2 Params

Coefficient

Formula Parameter

ImpactDirect contribution

ConstraintReal numbers

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.

Why the derivative of cos(ax) is −a·sin(ax)

Use the chain rule with the inner function u = ax, so du/dx = a.

d/dx[cos(u)] = −sin(u) · du/dx = −sin(ax) · a = −a·sin(ax).

Examples

d/dx[cos(3x)] = −3·sin(3x)

d/dx[cos(x/2)] = −½·sin(x/2)

d/dx[cos(πx)] = −π·sin(πx)

📖 Beginner's guideDerivative of cos(x) and cos(ax), with ExamplesThe derivative of cos(ax) is −a·sin(ax), and of cos(x) it is −sin(x). See why, work through examples like cos(3x), and try a free calculator with steps.

Worked Example: Derivative of cos(ax)

Given:

a = 1
x = 0

d/dx[cos(x)] = -sin(x)
At x=0: -sin(0) = 0

Answer: 0

Frequently Asked Questions

What is the Derivative of cos(ax) formula?

d/dx[cos(ax)] = −a·sin(ax). Compute the derivative of cos(ax) at a specific x value (x in radians).

How do I calculate Derivative of cos(ax)?

Enter Coefficient (a), Value of x (radians) (x) into the calculator. It applies d/dx[cos(ax)] = −a·sin(ax) and shows every step of the working.

What is the derivative of cos(ax)?

−a·sin(ax), by the chain rule.

What is the second derivative of cos(ax)?

−a²·cos(ax). Differentiating −a·sin(ax) gives −a·a·cos(ax).

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))/h Chain 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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Derivative of cos(ax)

Interactive Sandbox

Variable Mechanics

Coefficient

Value of x (radians)

At a Glance

Formula Nature & Mechanics

Common Mistakes & Traps

Dimensional Analysis & Units

Real-World & Industry Application

Why the derivative of cos(ax) is −a·sin(ax)

Examples

Worked Example: Derivative of cos(ax)

Frequently Asked Questions

What is the Derivative of cos(ax) formula?

How do I calculate Derivative of cos(ax)?

What is the derivative of cos(ax)?

What is the second derivative of cos(ax)?

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