# Rules of Differentiation

*The ultimate reason of things must lie in a necessary substance, in which the differentiation of thechanges only exists eminently as in their source; and this is what we call God.*(Leibniz)*The freedom of thought is a sacred right of every individual man, and diversity will continue to increase with the progress, refinement, and differentiation of the human intellect.*(Felix Adler)*How did Biot arrive at the partial differential equation? [the heat conduction equation] . . .Perhaps Laplace gave Biot the equation and left him to sink or swim for a few years in trying to derive it. That would have been merely an instance of the way great mathematicians since the very beginnings of mathematical research have effortlessly maintained their superiority over ordinary mortals. (*Clifford Truesdell)- Common integration is only the memory of differentiation. (Augustus De Morgan)

# Derivative of a Linear Combination

We have directly from the definition

and if is a constant

- .

Combining these results we have

where and are constants.

# Derivative of Product

If and are real-valued differentiable functions of , then

or

- ,

because

- .

# Derivative of a Quotient

We compute the derivative of the quotient assuming is differentiable:

up to a quadratic deviation in . Thus, by combination with the previous result,

- .

# The Chain Rule

If and are two differentiable functions, then the composite function is differentiable with derivative

- }.

To see this we estimate

up to terms of order .

# Read More

- \hyperref[diffrules]{Rules of Differentiation}
- \hyperref[intprops]{Rules of Integration}
- Differentiation Rules

# To Think About

- How sensitive is differentiation to perturbations of function values?
- How to compute thye derivate of a function if analytic differentiation is not an option, because it is too difficult or the function is not given by an analytic expression?

# Watch

- A no-where differentiable Weierstrass function
- Weierstrass rap
- Perelman million dollar Poincare rap
- Math professor

Leibniz’s first paper on calculus, Acta Eruditorum, 1684, with the above rules for differentiation.

Next: Proof of Fundamental Theorem Previous: Derivative with respect to Space

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