Mathematics · Calculus
Trigonometric Derivative Calculator
Differentiate A sin(kx+φ) and evaluate both the function and derivative at x.
Inputs and results stay in this browser. Change one value at a time to explore the relationship.
Calculation steps
- Inner phase=3×0+0=0.
- Multiply by the inner derivative 3: f′(0)=2×3cos(0)=6.
- The original function value is 0.
Understand Trigonometric derivative
One idea, three depths
Choose how deeply to explain Trigonometric derivative
Trigonometric derivative: Differentiate A sin(kx+φ) and evaluate both the function and derivative at x.
Age 5 Explain it to a 5-year-old Start with a picture
Think of watching something move or grow. Calculus helps measure tiny changes and how they add together. For example: For 2sin(3x) at x=0, the derivative is 6. The answer tells you Derivative at x.
Age 15 Explain it to a 15-year-old Connect it to the formula
Differentiating sine produces cosine, while the chain rule contributes the inner rate k. This pattern drives wave mechanics. The rule is d[A sin(kx+φ)]/dx=Ak cos(kx+φ). Its input values are Amplitude A, Angular coefficient k, Phase φ in radians, Evaluation point x, and the main result is Derivative at x. For example: For 2sin(3x) at x=0, the derivative is 6.
College Explain it at college level State the model precisely
This calculator evaluates a calculus model over the stated real-valued domain. The implemented relation is d[A sin(kx+φ)]/dx=Ak cos(kx+φ), evaluated from Amplitude A, Angular coefficient k, Phase φ in radians, Evaluation point x to produce Derivative at x. Differentiating sine produces cosine, while the chain rule contributes the inner rate k. This pattern drives wave mechanics. This compact symbolic model covers the displayed function family, not every function, discontinuity or domain restriction. Do not omit the inner coefficient k, and enter x and phase in radians.
Inputs and valid domain
- Amplitude A must be a finite real value.
- Angular coefficient k must be a finite real value.
- Phase φ in radians must be a finite real value.
- Evaluation point x must be a finite real value.
Important boundary: Do not omit the inner coefficient k, and enter x and phase in radians.
The formula
d[A sin(kx+φ)]/dx=Ak cos(kx+φ)
How the calculator works through it
It substitutes Amplitude A, Angular coefficient k, Phase φ in radians, Evaluation point x into the formula and exposes every numerical step above. The main output is Derivative at x, accompanied by Function value, Phase kx+φ.
Read the result correctly
The Derivative at x is the direct answer to “differentiate a sin(kx+φ) and evaluate both the function and derivative at x.” Read it with the units shown beside the inputs; a sign, angle, percentage or rate changes what the number means.
A worked check
For 2sin(3x) at x=0, the derivative is 6.
Where this model stops being reliable
This compact symbolic model covers the displayed function family, not every function, discontinuity or domain restriction. In particular, do not omit the inner coefficient k, and enter x and phase in radians.
Learn it by changing one value
Begin with the worked example, then change one value while keeping the others fixed. Compare the new result and calculation steps to identify which part of the formula changed.
Dictionary terms behind this calculator
Continue with a free textbook
OpenStax reading and academic references
Use the calculator as the worked interaction, then continue into the peer-reviewed textbook context. MW SysArc links to OpenStax; the explanation on this page is original and does not reproduce the book.
Calculus Volume 1
Read OpenStax Calculus: Derivatives and integrationCite this book
- APA 7
- Strang, G., & Herman, E. (2016). Calculus volume 1. OpenStax. https://openstax.org/books/calculus-volume-1/pages/1-introduction
- MLA 9
- Strang, Gilbert, and Edwin Herman. Calculus Volume 1. OpenStax, 2016, https://openstax.org/books/calculus-volume-1/pages/1-introduction.
- Chicago author-date
- Strang, Gilbert, and Edwin Herman. 2016. Calculus Volume 1. Houston, TX: OpenStax. https://openstax.org/books/calculus-volume-1/pages/1-introduction.
OpenStax books are free to read online. Their current reuse licence is CC BY-NC-SA; follow the licence shown on the linked book before redistributing or adapting its content.
Clear answers
Frequently asked questions
What does the Trigonometric derivative do?
Differentiate A sin(kx+φ) and evaluate both the function and derivative at x.
How does the Trigonometric derivative work?
The calculator applies d[A sin(kx+φ)]/dx=Ak cos(kx+φ). Differentiating sine produces cosine, while the chain rule contributes the inner rate k. This pattern drives wave mechanics.
What can I learn from the Trigonometric derivative?
It connects the mathematical rule to your chosen numbers and shows each calculation step. Change one input at a time to see how the result responds.
Does MW SysArc receive or store what I enter?
No. The calculation runs locally in your browser. MW SysArc does not receive or store your calculation inputs.
How should I use the result?
Use the steps to understand the method, then verify important school or professional work using the notation and rounding rules required in your setting.
Mathematics → algorithm → program
Implement this calculation in code
These are direct reference implementations of the calculator's principal relationship and first output. They run locally and include a small known-answer check where the language supports it.
Algorithm
- Read Amplitude A, Angular coefficient k, Phase φ in radians, Evaluation point x.
- Evaluate the principal relationship: d[A sin(kx+φ)]/dx=Ak cos(kx+φ).
- Return Derivative at x and check the domain conditions described above.
Python
from math import *
def sinusoidal_derivative(a, b, c, x) -> float:
return ((a * b) * cos(((b * x) + c)))
assert abs(sinusoidal_derivative(2, 3, 0, 0) - 6) < 1e-6 * max(1.0, abs(6))
C
#include <assert.h>
#include <math.h>
double sinusoidal_derivative(double a, double b, double c, double x) {
return ((a * b) * cos(((b * x) + c)));
}
int main(void) {
const double expected = 6;
const double actual = sinusoidal_derivative(2, 3, 0, 0);
assert(fabs(actual - expected) < 1e-6 * fmax(1.0, fabs(expected)));
}
C++
#include <cassert>
#include <cmath>
#include <numbers>
double sinusoidal_derivative(double a, double b, double c, double x) {
return ((a * b) * std::cos(((b * x) + c)));
}
int main() {
constexpr double expected = 6;
const double actual = sinusoidal_derivative(2, 3, 0, 0);
assert(std::fabs(actual - expected) < 1e-6 * std::fmax(1.0, std::fabs(expected)));
}
Linux x86-64 assembly
x86-64 NASM · System V ABI · Linux · SSE2 with libm where required
; double sinusoidal_derivative(double a, double b, double c, double x)
; Linux x86-64 NASM · System V ABI · first eight doubles in xmm0–xmm7
extern cos
global sinusoidal_derivative
section .text
sinusoidal_derivative:
push rbp
mov rbp, rsp
sub rsp, 80
movsd [rbp-8], xmm0
movsd [rbp-16], xmm1
movsd [rbp-24], xmm2
movsd [rbp-32], xmm3
movsd xmm0, [rbp-8]
mulsd xmm0, [rbp-16]
movsd [rbp-48], xmm0
movsd xmm0, [rbp-16]
mulsd xmm0, [rbp-32]
movsd [rbp-72], xmm0
movsd xmm0, [rbp-72]
addsd xmm0, [rbp-24]
movsd [rbp-64], xmm0
movsd xmm0, [rbp-64]
call cos wrt ..plt
movsd [rbp-56], xmm0
movsd xmm0, [rbp-48]
mulsd xmm0, [rbp-56]
movsd [rbp-40], xmm0
movsd xmm0, [rbp-40]
leave
ret
Floating-point answers can differ slightly by language, compiler and processor. Compare within a suitable tolerance rather than assuming every decimal representation will be identical.
Last reviewed 2026-07-21. Calculations tested 2026-07-21.