Trigonometric Derivative Calculator

Calculator inputs

Use the responsive form below. It shows three columns on large screens, two on medium screens, and one on mobile.

Amplitude (a)

Outside multiplier for the full expression.

Trigonometric function

Choose one of the six standard trig functions.

Power (n)

Positive integer power applied to the trig function.

Inner coefficient (b)

Controls horizontal scaling and chain-rule impact.

Inner constant (c)

Adds a phase shift inside the function.

Evaluate at x

Optional numeric point for f(x) and f′(x).

Graph minimum x

Default covers about −2π in radians.

Graph maximum x

Default covers about +2π in radians.

Graph points

Higher values create smoother graph lines.

Reset

Example data table

Example	Input expression	Derivative result	Main rule used
1	sin(x)	cos(x)	Basic derivative rule
2	2cos(3x)	-6sin(3x)	Chain rule
3	4tan²(x - 1)	8tan(x - 1)sec²(x - 1)	Power rule and chain rule
4	-3sec(2x + 0.5236)	-6sec(2x + 0.5236)tan(2x + 0.5236)	Product-style trig derivative pattern
5	5csc³(0.5x)	-7.5csc³(0.5x)cot(0.5x)	Power rule with reciprocal trig derivative

Formula used

General model: f(x) = a[g(x)]ⁿ, where g(x) = trig(bx + c).
Power rule form: d/dx[a(g(x))ⁿ] = a·n(g(x))^n-1·g′(x).
Chain rule for the inside term: d/dx(bx + c) = b.
Base derivatives: d/dx[sin(u)] = cos(u), d/dx[cos(u)] = -sin(u), d/dx[tan(u)] = sec²(u).
Reciprocal derivatives: d/dx[sec(u)] = sec(u)tan(u), d/dx[csc(u)] = -csc(u)cot(u), d/dx[cot(u)] = -csc²(u).
For this page, all x-values are treated in radians.

How to use this calculator

Enter the outside coefficient a, choose a trig function, and set the integer power n.
Enter the inside expression values b and c for the linear angle bx + c.
Optionally enter an x-value to compute both the original function and its derivative at one point.
Choose graph limits and the number of plotting points for the visual comparison of both curves.
Click Calculate Derivative to show the symbolic derivative above the form, then export with CSV or PDF if needed.

FAQs

1) What type of trig expression does this calculator support?

It handles expressions built as a multiplied trig power: a·trig(bx+c)ⁿ. That covers many classroom problems involving scaling, shifting, and repeated trig factors inside a single expression.

2) Does it use degrees or radians?

The calculator uses radians. That matches standard calculus derivative rules. If your problem is written in degrees, convert the angle measure first or rewrite the expression in radians before evaluating points.

3) Why does the derivative become undefined at some x-values?

Functions such as tan, sec, csc, and cot have restricted points where division by zero appears. At those x-values, the original function, derivative, or both may be undefined, so the calculator marks them clearly.

4) Does the calculator apply the chain rule automatically?

Yes. The inner linear term bx+c is differentiated as b, and that factor is multiplied into the final derivative automatically. This is especially important whenever the inside coefficient is not 1.

5) Can I use it for powers like sin⁴(3x)?

Yes. Set the function to sin, enter the inside coefficient 3, keep the constant as needed, and choose power 4. The calculator will apply the power rule and chain rule together.

6) What does the graph show?

The graph plots the original expression and its derivative over your selected x-range. This helps you compare turning behavior, steepness, oscillation changes, and points where the derivative crosses zero.

7) What is included in the CSV and PDF exports?

Both exports include the main inputs, symbolic result, graph range, and the worked derivative steps. They are designed for revision notes, classroom sharing, and quick record keeping.

8) Can I extend this page for more advanced calculus features?

Yes. You can expand it to support second derivatives, inverse trig rules, product rule expressions, or symbolic simplification. The current structure is already organized to make those upgrades straightforward.