A sample derivative by the definition: f(x) = 4/(2x+3)

The official definition (just look at the blue stuff):

> Limit((f(x+h)-f(x))/h,h=0);

Some key steps:

Defining a function in Maple:

> f:= x -> 4/(2*x+3);

Compute f(x+h) and f(x):

> `f(x+h)`=f(x+h);
`f(x)`=f(x);

Subtracting those quantities and getting a common denominator yields:

> `f(x+h) - f(x)`=simplify(f(x+h)-f(x));

Now, divide by h and cancel:

> `f(x+h) - f(x)`/h=simplify((f(x+h)-f(x))/h);

Let h go to zero:

> subs({h=0},simplify((f(x+h)-f(x))/h));

Or, all at once:

> Limit((`f(x+h)-f(x)`)/h,h=0)=limit((f(x+h)-f(x))/h,h=0);

Now, erase the definition of f(x):

> f:='f';

Maple doesn't know the "prime notation" for derivatives, so these are just answers.
 

• f '(x) / f(x)

• h'(x) = g'(u(x) )* u'(x), so h'(5) = g'(u(5) )*u'(5) = g'(1)*(-2) = 7*(-2) = -14

• h'(3^x)*ln(3)*3^x + 10g'(5x)

• f '(ln(x) ) / x

> diff(sin(2*theta),theta);

> diff(3^w+sin(3*Pi),w);

> diff(sin(3*t)^2,t);

> diff(x^(-2/3)*sin(x),x);

> diff(sqrt(376),x);

> diff((3*x+2)^(1/3),x);

> diff(4*t^(-1/3)+5/(t^2)-sqrt(3*t),t);

> diff(4^(3*t-2),t);

> diff(5*w^7-3*w+12,w);

> diff((3*x^2-4)^4,x);

> diff(x*sin(x),x);

> diff(4*z^5-2*z^3+9,z);

> diff(3/ln(x),x);

> diff(exp(3*x),x);

> diff(x*ln(4*x)+5,x);

> diff(ln(4*x^2+12),x);

> diff(theta*tan(theta),theta);

> diff(ln(3^x),x);

> diff(124^x,x);

> diff(ln(3*t),t);

> diff(sqrt(3*x+5)-1/(3^x),x);

> diff((2*x-3)/(7+4*x),x);
`same as:`=factor(%);

> diff(exp(x)+x^exp(1)-Pi^2,x);

> diff( (x+cos(x))^(-3),x);

> diff(exp(3*x)*cos(-2*x),x);

> diff(5^sqrt(x),x);

> diff(sin(5-t),t);

> diff(x^3*exp(2*x),x);

> diff(x*cos(3*x-2),x);

> diff(exp(sin(x)),x);

> diff(arcsin(2*theta),theta);

> diff(arctan(w),w);

> diff(cos(Pi*t+Pi/4),t);

> diff(2-tan(4*x),x);
`same as:`=-4*sec(4*x)^2;

> diff(x/sin(x),x);

> diff(5/(1-t),t);

> diff(cos(1-theta^2),theta);

> diff(cos(2*z)/12,z);

> diff(t^3*ln(t),t);

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