Differentiate common expressions quickly. View slopes, graphs, tables, and exportable results. Practice rules using guided inputs and clear examples.
For sine, cosine, tangent, exponential, and logarithmic templates, the power field is not used in the derivative formula.
| Function | Derivative | At x = 1 |
|---|---|---|
| 3(2x + 1)^4 | 24(2x + 1)^3 | 648 |
| 5sin(3x) | 15cos(3x) | -14.849887 |
| 2e^(x - 1) | 2e^(x - 1) | 2 |
| 7ln(4x + 2) | 28/(4x + 2) | 4.666667 |
Power rule: d/dx [a(bx + c)^n] = a·n·b(bx + c)^(n − 1)
Sine rule: d/dx [a sin(bx + c)] = a·b cos(bx + c)
Cosine rule: d/dx [a cos(bx + c)] = −a·b sin(bx + c)
Tangent rule: d/dx [a tan(bx + c)] = a·b sec²(bx + c)
Exponential rule: d/dx [a e^(bx + c)] = a·b e^(bx + c)
Log rule: d/dx [a ln(bx + c)] = a·b / (bx + c)
Numeric check: f′(x) ≈ [f(x + h) − f(x − h)] / 2h, where h is very small.
Derivatives measure change. They show how fast a function moves. They also show direction. A positive derivative means growth. A negative derivative means decline. A zero derivative can signal a turning point. This makes derivatives useful in maths, science, and engineering.
This calculator handles common function templates. You can differentiate power, sine, cosine, tangent, exponential, and logarithmic forms. It also evaluates the function and derivative at a chosen x-value. That gives both the rule and the exact slope at one point.
The page uses analytical rules first. Then it performs a numeric check. This second method estimates the slope from nearby points. Matching values increase confidence in the answer. Small differences can happen because rounded numbers are shown on screen.
The graph compares the original function with its derivative. This view makes slope behavior easier to understand. You can spot peaks, valleys, steep sections, and flat regions. It also helps students connect formulas to visual changes.
Use the power template for expressions like a(bx + c)^n. Use trigonometric templates for wave problems. Use the exponential template for growth models. Use the logarithmic template for rate analysis. If your range crosses an invalid domain, adjust the graph limits and test again.
Students can use it for homework practice. Teachers can use it for classroom demos. Engineers can use it to inspect rate changes quickly. Anyone learning calculus can use it to compare symbolic rules with numeric slope estimates and graph behavior.
The derivative represents the rate of change of a function. It tells you how quickly the output changes when x changes slightly. It also gives the slope of the tangent line at a chosen point.
No. This version focuses on common templates such as power, trig, exponential, and logarithmic forms. It is designed for fast and clear results instead of a full symbolic algebra engine.
The numeric check estimates the slope using nearby values. It helps confirm the analytical derivative. If both values are close, the result is usually reliable. Small display rounding can still cause slight differences.
Some functions are not valid for all x-values. For example, ln(bx + c) requires a positive inner value. Tangent becomes undefined where cosine equals zero. Adjust the input or graph range to continue.
The power field is used for the template a(bx + c)^n. It is not used for sine, cosine, tangent, exponential, or logarithmic templates. Those functions follow their own derivative rules.
The graph shows the original function and its derivative together. This makes it easier to see where the function rises, falls, or flattens. It also helps connect formulas with shape changes.
Yes. You can export the generated table as CSV. You can also create a PDF snapshot of the result section. These exports are useful for reports, notes, and quick sharing.
This calculator is useful for students, teachers, tutors, and technical users. It works well for calculus practice, concept review, slope checking, and quick demonstrations of standard differentiation rules.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.