2.3 Differentiation

The core derivatives are d(sinh x)/dx=cosh x, d(cosh x)/dx=sinh x and d(tanh x)/dx=sech²x. Their integrals follow by reversing these relationships, with a constant of integration.

Use the chain rule for sinh(ax+b), and remember that ∫sech²x dx=tanh x. For logarithmic forms, check the domain and absolute-value convention where relevant.

∫3sinh(3x+1)dx=cosh(3x+1)+C, because the inner derivative 3 cancels the coefficient.

The derivative of cosh x is +sinh x, not −sinh x, and tanh is not differentiated like ordinary tan x.

When an integral contains a combination such as sinh x and cosh x, set u to a useful hyperbolic expression or use t=e^x. The identities cosh²x−sinh²x=1 and dx=dt/t can reduce the problem to algebraic or logarithmic terms.

Choose a substitution that matches the differential factor; do not expand blindly. Back-substitute and differentiate the result to verify the antiderivative.

With t=e^x, sinh x=(t−t⁻¹)/2 and cosh x=(t+t⁻¹)/2, so an integral in sinh and cosh becomes a rational expression in t.

A substitution is not complete until dx and the limits, if definite, are transformed consistently.

The Maclaurin series of f is f(0)+f′(0)x+f″(0)x²/2!+… . It is the Taylor expansion about x=0 and is useful when the series converges and a finite truncation gives the required accuracy.

Find a pattern in derivatives or use known expansions, state the interval or radius of convergence when relevant, and estimate the remainder if an error bound is required.

e^x=1+x+x²/2!+x³/3!+…; for small x, retaining terms through x² gives a controlled approximation whose error is of the order of the next term.

A formal power series is not automatically valid for every x, and truncating after a convenient term does not prove a numerical accuracy.