Derivatives
Sum, product, and quotient rules
Combine known derivatives without expanding every expression. Track which terms depend on the input.
1. Constant multiples
1 / 4f(x)=3x^2,\quad f\prime(x)=6x A constant multiplier scales every tangent slope: (cf)′=cf′. For 3x², the derivative is 6x. An added constant has derivative zero.
- f(a)
- 3
- f′(a)
- 6
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. The sum rule
2 / 4f(x)=x^2+\sin x,\quad f\prime(x)=2x+\cos x For differentiable u and v, (u+v)′=u′+v′. Thus x²+sin(x) has derivative 2x+cos(x), using radians.
- 2a
- 2
- cos(a)
- 0.540302
- f′(a)
- 2.540302
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. The product rule
3 / 4f=x^2\sin x,\quad f\prime=2x\sin x+x^2\cos x Both factors change in x²sin(x). The derivative is u′v+uv′, not u′v′. One term accounts for each changing factor.
- u′v
- 1.682942
- uv′
- 0.540302
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. The quotient rule
4 / 4f=\frac{x^2}{x+1},\quad f\prime=\frac{x(x+2)}{(x+1)^2} For v≠0, (u/v)′=(u′v−uv′)/v². For x²/(x+1), the result is x(x+2)/(x+1)². The excluded input −1 remains excluded.
- f(a)
- 0.5
- f′(a)
- 0.75
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.