Derivatives
The chain rule
A composition changes through an inner input. Multiply the outer derivative, evaluated at the inner function, by the inner derivative.
1. A square of a linear function
1 / 4f=(2x+1)^2,\quad f\prime=4(2x+1) Let u=2x+1 and F(u)=u². Then F′(u)=2u and u′=2, so the derivative is 4(2x+1).
- u(a)
- 1
- u′(a)
- 2
- f′(a)
- 4
The solid curve is f; the dashed line is its tangent at a. Read f(a) as height and f′(a) as slope. Questions use the input stated in their text.
2. Sine of a square
2 / 4f=\sin(x^2),\quad f\prime=2x\cos(x^2) For sin(x²), the outer derivative is cos(x²) and the inner derivative is 2x. Work in radians.
- u(a)
- 1
- f′(a)
- 1.080605
The solid curve is f; the dashed line is its tangent at a. Read f(a) as height and f′(a) as slope. Questions use the input stated in their text.
3. An exponential composition
3 / 4f=e^{-x^2},\quad f\prime=-2xe^{-x^2} For e^(−x²), the exponential stays while the inner derivative contributes −2x. This explains the positive slopes on the left and negative slopes on the right.
- f(a)
- 0.367879
- f′(a)
- -0.735759
The solid curve is f; the dashed line is its tangent at a. Read f(a) as height and f′(a) as slope. Questions use the input stated in their text.
4. A logarithmic composition
4 / 4f=\ln(1+x^2),\quad f\prime=\frac{2x}{1+x^2} Because 1+x² is always positive, ln(1+x²) is defined for every real x. Its derivative is the inner derivative divided by the inner function.
- 1+a²
- 2
- f′(a)
- 1
The solid curve is f; the dashed line is its tangent at a. Read f(a) as height and f′(a) as slope. Questions use the input stated in their text.