DPST1013 Chap.10 Logarithmic and Hyperbolic Functions
Logarithmic and Hyperbolic Functions
Logarithmic, exponential and hyperbolic functions extend the standard differentiation and integration toolkit while reinforcing the importance of domains and inverse mappings. The natural logarithm is defined through an integral of the reciprocal function, so its derivative follows from the fundamental theorem of calculus. Its inverse is the exponential function.
This relationship explains their complementary domains and ranges and provides a reliable route to derivative formulas for other positive bases. Logarithms also reorganise expressions. Products become sums, quotients become differences, and changing powers become products involving an exponent and a logarithm. Those transformations are useful only where the logarithms are defined.
For a positive function raised to a changing exponent, logarithmic differentiation separates the effects of the changing base and exponent. The derivative of the logarithm must then be multiplied by the original function to recover the requested derivative. Hyperbolic sine and cosine are built from exponential sums and differences.
Their definitions explain their symmetry, their derivatives and the identity involving the difference of their squares. Similar notation to circular trigonometric functions should not encourage copying signs blindly. In particular, the derivative of hyperbolic cosine is positive hyperbolic sine. Hyperbolic tangent is their ratio and has a restricted range even though its real domain is unrestricted.
Inverse hyperbolic functions connect these ideas again: they can be expressed using logarithms and provide primitives for useful radical integrals. Choosing a hyperbolic substitution can turn a square-root expression into a simple identity, provided the parameter interval and sign are handled correctly. Across the chapter, return repeatedly to the defining relationships rather than memorising disconnected formulas.
Exact checks at zero or logarithms of small positive integers help verify both calculations and interpretations.
What this chapter covers
- 01
Natural logarithm and exponential: use the integral definition to obtain the reciprocal derivative, then treat exponential as the inverse. Keep positive logarithm inputs distinct from the unrestricted real inputs allowed by the natural exponential.
- 02
Other bases and logarithmic integration: rewrite positive-base powers with the natural exponential and use the change-of-base formula for logarithms. The primitive of one over x is a logarithm of an absolute value on an interval avoiding zero.
- 03
Logarithmic differentiation and power limits: taking logarithms separates a changing exponent from a changing base. For limits, calculate the logarithmic limit and then return through the continuous exponential function rather than stopping at the transformed result.
- 04
Hyperbolic definitions and identities: derive values from half-sums and half-differences of exponentials. Check even and odd symmetry, the difference-of-squares identity, and derivative signs before using composite arguments.
- 05
Inverse hyperbolic functions and substitution: identify the required inverse domain and branch. Match a radical to an appropriate identity, transform the differential and bounds, and confirm the result by differentiating the inverse-function expression.
A radical integral through hyperbolic sine
- 2Set x=3sinh u, so dx=3cosh u du. The radical becomes 3√(1+sinh²u)=3cosh u. The last equality uses the identity and the fact that hyperbolic cosine is positive, rather than cancelling an unchecked square root.
- 2The lower endpoint x=0 corresponds to u=0, while x=3 corresponds to sinh u=1 and hence u=arsinh 1. The differential and denominator cancel exactly, leaving the integral of one from zero to arsinh 1.
- 2Thus the result is arsinh 1=ln(1+√2), approximately 0.881374. Independently, differentiating arsinh(x/3) gives (1/3)/√(1+x²/9)=1/√(x²+9), since the positive scale three can be taken outside the radical. Endpoint evaluation confirms the same exact value.
Key terms
- Natural logarithm
- The integral of the reciprocal function from one to a positive input; it is also the inverse of the natural exponential.
- Logarithmic derivative
- The quotient f′/f, obtained by differentiating ln f where f is positive, or ln|f| on an interval where f is nonzero.
- Changing exponent
- An exponent depending on the input, whose derivative contributes an additional term when a power is differentiated logarithmically.
- Hyperbolic sine
- Half the difference between a natural exponential and its reciprocal exponential; an odd function increasing across the real line.
- Hyperbolic cosine
- Half the sum of opposite exponentials; an even, positive function whose minimum value is one.
- Inverse hyperbolic sine
- The inverse of hyperbolic sine on the real line, expressible as a logarithm and useful for integrating reciprocal square-root sums.
- Hyperbolic substitution
- A change of variable using a hyperbolic function so that a radical simplifies through a hyperbolic identity.
Logarithmic and Hyperbolic Functions FAQ
Why does a logarithm need a positive real argument?
The natural logarithm is the inverse of an exponential whose real outputs are all positive. Its integral definition likewise uses a positive endpoint connected to one without crossing the reciprocal singularity at zero.
When does the ordinary power rule fail?
The usual power rule assumes its exponent is constant. If both base and exponent depend on the input, logarithmic differentiation accounts for both changes. Treating either quantity as constant omits a required contribution.
Why is hyperbolic cosine positive everywhere?
It is half the sum of two positive exponentials. Consequently it never vanishes, and a square root of its square equals the function itself rather than requiring an unresolved sign choice.
Is inverse hyperbolic cosine defined for negative inputs?
Its standard real domain begins at one because it reverses hyperbolic cosine on the nonnegative branch. A negative input lies outside that real range, even though other hyperbolic inverses may accept all real inputs.
Why exponentiate after a logarithmic limit?
Taking a logarithm changes the expression whose limit is being computed. If that transformed expression approaches a finite number, continuity of the exponential returns the limit of the original positive expression. The two numerical answers usually differ.
Exam move
Build a small derivation chain rather than a disconnected formula list. Start with the reciprocal integral definition of logarithm, identify exponential as its inverse, and derive the two hyperbolic functions from exponentials. Verify values at zero and at the logarithm of a small integer. These exact checks make sign and reciprocal mistakes visible before you tackle longer expressions.
Next mix three problem types: a changing power requiring logarithmic differentiation, a power limit requiring a logarithmic transformation, and a radical integral requiring a hyperbolic substitution. Write the domain beside the first line of each solution. After finishing, check a derivative by an alternative representation or check a primitive by differentiation. For inverse functions, name the branch and range explicitly.
Keep exact logarithmic answers before decimal evaluation, and explain why every square-root simplification respects the sign of the expression inside it.
Working through Logarithmic and Hyperbolic Functions in DPST1013? Sia is AskSia’s AI Mathematics tutor — ask any DPST1013 Logarithmic and Hyperbolic Functions question and get a clear, step-by-step explanation grounded in how DPST1013 is taught and assessed. Read this chapter free, then take your hardest questions to Sia.
