Calculator Form
Formula Used
The calculator uses the Fundamental Theorem of Calculus with the chain rule. The advanced model is:
g(x) = ∫ from l(x) to u(x) of k·t·sin(t) dt
g′(x) = k·u(x)·sin(u(x))·u′(x) − k·l(x)·sin(l(x))·l′(x)
For the common case g(x)=∫ from a to x of t·sin(t) dt, the derivative is g′(x)=x·sin(x).
The antiderivative used for the accumulated value is ∫ t·sin(t)dt = sin(t) − t·cos(t).
How to Use This Calculator
- Enter the value of x.
- Keep coefficient k as 1 for the basic function.
- Use upper slope 1 and upper intercept 0 for upper limit x.
- Use lower slope 0 for a constant lower limit.
- Enter the lower intercept as the constant lower limit.
- Select decimal places for rounding.
- Press the calculate button.
- Download CSV or PDF when the result appears.
Example Data Table
| x | k | Upper m | Upper b | Lower n | Lower c | Derivative |
|---|---|---|---|---|---|---|
| 1.00 | 1.00 | 1.00 | 0.00 | 0.00 | 0.00 | 0.841471 |
| 2.00 | 1.00 | 1.00 | 0.00 | 0.00 | 0.00 | 1.818595 |
| 3.00 | 1.00 | 1.00 | 0.00 | 0.00 | 0.00 | 0.423360 |
| 2.00 | 2.00 | 1.00 | 0.00 | 0.00 | 0.00 | 3.637190 |
| 2.00 | 1.00 | 2.00 | 1.00 | 0.00 | 0.00 | -9.589243 |
| 2.00 | 1.00 | 1.00 | 0.00 | 1.00 | 0.00 | 0.000000 |
Understanding This Derivative Calculator
This calculator helps you study a common calculus task. It finds the derivative of an integral function. The default model is g(x) equal to the integral of t sin t with respect to t. The upper limit is usually x. The lower limit can be constant or moving.
Why the Result Matters
This type of problem appears in lessons on the Fundamental Theorem of Calculus. The theorem links accumulation and instantaneous change. An integral measures accumulated area. A derivative measures how that accumulated value changes. When the upper limit is x, the derivative becomes the integrand evaluated at x. For t sin t, the result is x sin x.
Advanced Input Control
The form also lets you use linear moving limits. You can set an upper limit as mx plus b. You can also set a lower limit as nx plus c. This is useful when limits are not simple. The chain rule is then needed. The calculator applies the upper contribution, then subtracts the lower contribution. It also multiplies each part by the derivative of its limit.
Numerical Value and Integral Value
The tool reports the derivative at the selected x value. It also reports the upper limit, lower limit, and accumulated integral value. The antiderivative of t sin t is sin t minus t cos t. With a coefficient, the whole expression is multiplied by that coefficient. Results are rounded by your selected precision.
Using Results in Study
You can compare several examples in the table. Try changing only one input at a time. This helps you see how each limit affects the derivative. The CSV export is useful for spreadsheets. The PDF export is useful for worksheets, reports, and class notes.
Practical Learning Notes
Always use radians for trigonometric calculus. Degree mode is not used here because derivative rules for sine and cosine assume radian measure. Review signs carefully when the lower limit moves. A moving lower limit is subtracted. This detail is a common source of mistakes. Use the step output to check every part before trusting the final answer.
Keep sample rows nearby, because repeated practice makes theorem patterns easier to remember and apply during homework checks or exams later.
FAQs
What does this calculator find?
It finds the derivative of an integral function based on t sin t. It can also handle coefficients and linear moving limits.
What is the default function?
The default model is g(x)=∫ from 0 to x of t sin(t) dt. Its derivative is x sin(x).
Does the calculator use radians?
Yes. Trigonometric derivative rules assume radians. Enter x and limit values as radian-based values for correct calculus results.
Can I use a moving upper limit?
Yes. Enter upper slope m and upper intercept b. The upper limit becomes u(x)=mx+b.
Can I use a moving lower limit?
Yes. Enter lower slope n and lower intercept c. The lower limit becomes l(x)=nx+c.
What formula is applied?
It applies g′(x)=k·u·sin(u)·u′−k·l·sin(l)·l′. This combines the theorem with the chain rule.
Why is the lower term subtracted?
A definite integral accumulates from the lower limit to the upper limit. A changing lower limit reduces or changes that accumulation.
Can I export my result?
Yes. After calculation, use the CSV or PDF button to save the displayed result table for later use.