Calculate derivatives step-by-step with symbolic differentiation
x^2, 2*x, x/2, sqrt(x), abs(x)sin(x), cos(x), tan(x), sec(x), asin(x)e^x, exp(x), 2^x, ln(x), log(x,10)sinh(x), cosh(x), tanh(x), asinh(x)
A derivative measures how a function changes as its input changes. Geometrically, the derivative at a point represents the slope of the tangent line to the function's graph at that point. In physics, derivatives represent rates of change: velocity is the derivative of position, and acceleration is the derivative of velocity.
The derivative of f(x) can be written in multiple ways:
If f(x) = xn, then f'(x) = n·xn-1
Example: d/dx[x³] = 3x²
If f(x) = g(x)·h(x), then f'(x) = g'(x)·h(x) + g(x)·h'(x)
Example: d/dx[x²·sin(x)] = 2x·sin(x) + x²·cos(x)
If f(x) = g(x)/h(x), then f'(x) = [g'(x)·h(x) - g(x)·h'(x)] / [h(x)]²
Example: d/dx[x/sin(x)] = [sin(x) - x·cos(x)] / sin²(x)
If f(x) = g(h(x)), then f'(x) = g'(h(x))·h'(x)
Example: d/dx[sin(x²)] = cos(x²)·2x
| Function | Derivative |
| sin(x) | cos(x) |
| cos(x) | -sin(x) |
| tan(x) | sec²(x) = 1/cos²(x) |
| ex | ex |
| ax | ax·ln(a) |
| ln(x) | 1/x |
| loga(x) | 1/(x·ln(a)) |
Find maximum and minimum values of functions. Critical points occur where f'(x) = 0 or f'(x) is undefined.
Velocity is the derivative of position: v(t) = ds/dt. Acceleration is the derivative of velocity: a(t) = dv/dt.
Determine where functions are increasing (f'(x) > 0) or decreasing (f'(x) < 0), and find inflection points using f''(x).
Marginal cost, marginal revenue, and marginal profit are all derivatives. They show how these quantities change with production level.
The second derivative f''(x) measures how the rate of change itself is changing (concavity):
In physics, the second derivative represents acceleration, while the third derivative (jerk) describes how acceleration changes over time.
Explore more calculus tools for complete mathematical analysis.
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