1 Derivative Definition and Rules

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1.1 Linear and Non-linear Functions

Linear functions have a constant slope, but what is the rate of change for a non-linear function as we shift along its domain?

1.2 Definition

(SB) Let \((x_0 ,f(x_0 ))\) be a point on te graph of \(y=f(x)\).

The derivative of \(f\) at \(x_0\) is the slope of the tangent line to the graph of \(f\) at \((x_0 ,f(x_0 ))\).

There are some common ways of denoting derivative of funtion \(f\) at \(x_0\):

  • \(\displaystyle f^{\prime } (x_0 )\)

  • \(\displaystyle \frac{df}{dx}(x_0 )\)

  • \(\displaystyle \frac{dy}{dx}(x_0 )\)

  • \(\displaystyle f_x (x_0 )\)

We write this analyticaly as:

\[f^{\prime } (x_0 )=\lim_{h\to 0} \frac{f(x_0 +h)-f(x_0 )}{h}\]

If this limit exists, then the function \(f\) is differentiable at \(x_0\).

We will use this formula to derive first order taylor approximation. And this will also appear when we derive the formula for point elasticity.

1.3 Derivative Rules–Constant Rule

given constant \(k\),:

  • \(\displaystyle f(x)=a\cdot x\)

  • \(\displaystyle f^{\prime } (x_0 )=a\)

syms x a
f(x, a) = a*x

f(x, a) =

\(\displaystyle a\,x\)

dfk = diff(f,x)

dfk(x, a) =

\(\displaystyle a\)

1.4 Derivative Rules–Power Rule (Polynomial Rule)

(SB) For any positive integer \(k\) (or real number \(k\)), the derivative of \(f(x)=x^k\) at \(x_0\) is:

  • \(\displaystyle f(x)=x^k\)

  • \(\displaystyle f^{\prime } (x_0 )=k\cdot x_0^{k-1}\)

syms x a k
f(x, a, k) = a*x^k

f(x, a, k) =

\(\displaystyle a\,x^k\)

dfk = diff(f,x)

dfk(x, a, k) =

\(\displaystyle a\,k\,x^{k-1}\)

1.5 Derivative Rules–Chain Rule

  • \(\displaystyle f(x)=p(q(x))\)

  • \(\displaystyle f^{\prime } (x_0 )=p^{\prime } (q(x_0 ))\cdot q^{\prime } (x_0 )\)

syms x a k
f(x, a, k) = (a*x)^k

f(x, a, k) =

\(\displaystyle {{\left(a\,x\right)}}^k\)

dfk = diff(f,x)

dfk(x, a, k) =

\(\displaystyle a\,k\,{{\left(a\,x\right)}}^{k-1}\)

1.6 Derivative Rules–Sum (and difference) Rule

Given functions \(p\) and \(q\) that are differentiable at \(x\), then:

  • \(\displaystyle f(x)=p(x)+q(x)\)

  • \(\displaystyle f^{\prime } (x)=p^{\prime } (x)+q^{\prime } (x)\)

syms x a b c d
f(x, a, b, c, d) = a*x^b + c*x^d

f(x, a, b, c, d) =

\(\displaystyle a\,x^b +c\,x^d\)

dfk = diff(f,x)

dfk(x, a, b, c, d) =

\(\displaystyle a\,b\,x^{b-1} +c\,d\,x^{d-1}\)

1.7 Derivative Rules–Product Rule

Given functions \(p\) and \(q\) that are differentiable at \(x\), then:

  • \(\displaystyle f(x)=p(x)\cdot q(x)\)

  • \(\displaystyle f^{\prime } (x)=p^{\prime } (x)\cdot q(x)+p(x)\cdot q^{\prime } (x)\)

syms x a b c d
f(x, a, b, c) = (a*x^b)*(c*x^d)

f(x, a, b, c) =

\(\displaystyle a\,c\,x^b \,x^d\)

dfk = diff(f,x)

dfk(x, a, b, c) =

\(\displaystyle a\,b\,c\,x^d \,x^{b-1} +a\,c\,d\,x^b \,x^{d-1}\)

1.8 Derivative Rules–Quotient Rule

Given functions \(p\) and \(q\) that are differentiable at \(x\), then:

  • \(\displaystyle f(x)=\frac{p(x)}{q(x)}\)

  • \(\displaystyle f^{\prime } (x)=\frac{p^{\prime } (x)\cdot q(x)-p(x)\cdot q^{\prime } (x)}{(q(x))^2 }\)

Note that the quotient rule is based on the product rule, because:

  • \(\displaystyle f(x)=\frac{p(x)}{q(x)}=p(x)\cdot \frac{1}{q(x)}\)

So you can derive the quotient rule formula based on the product rule where the first term is \(p(x)\) and the second term is \(\frac{1}{q(x)}\).

syms x a b c d
f(x, a, b, c) = (a*x^b)/(c*x^d)

f(x, a, b, c) =

\(\displaystyle \frac{a\,x^b }{c\,x^d }\)

dfk = diff(f,x)

dfk(x, a, b, c) =

\(\displaystyle \frac{a\,b\,x^{b-1} }{c\,x^d }-\frac{a\,d\,x^b }{c\,x^{d+1} }\)

1.9 Derivative Rules–Exponential

We use exponential functions in economnics a lot:

  • \(\displaystyle f(x)=\exp (a\cdot x)\)

  • \(\displaystyle f^{\prime } (x)=a\cdot \exp (a\cdot x)\)

syms x a
f(x, a) = exp(a*x)

f(x, a) =

\(\displaystyle {\mathrm{e}}^{a\,x}\)

dfk = diff(f,x)

dfk(x, a) =

\(\displaystyle a\,{\mathrm{e}}^{a\,x}\)

This is a special case of any power function

  • \(\displaystyle f(x)=c^{a\cdot x}\)

  • \(\displaystyle f^{\prime } (x)=a\cdot (\log c)\cdot c^{a\cdot x}\)

note that \(log(exp(c))=c\)

syms x a c
f(x, a, c) = c^(a*x)

f(x, a, c) =

\(\displaystyle c^{a\,x}\)

dfk = diff(f,x)

dfk(x, a, c) =

\(\displaystyle a\,c^{a\,x} \,\log \left(c\right)\)

1.10 Derivative Rules–Log

We use Log functions in economnics a lot:

  • \(\displaystyle f(x)=\log (a\cdot x)\)

  • \(\displaystyle f^{\prime } (x)=\frac{1}{x}\)

note that the c cancels out.

syms x a
f(x, a) = log(a*x)

f(x, a) =

\(\displaystyle \log \left(a\,x\right)\)

dfk = diff(f,x)

dfk(x, a) =

\(\displaystyle \frac{1}{x}\)