MAT148H1 Chap.7 Trigonometric, Exponential and Log Derivatives
Trigonometric, Exponential and Log Derivatives
Where these derivatives sit in the course
Unit 4 of the syllabus introduces transcendental functions: sine, cosine and tangent, their inverses, the exponential eˣ and the natural logarithm. These functions cannot be built from polynomials by finitely many algebraic steps, so each derivative has to be earned from a limit, from an earlier rule, or from an inverse relationship.
In a proof based course the derivation counts as much as the formula. A student who can rebuild d/dx tan x from the quotient rule in four lines rarely needs to memorise the full table of six trigonometric derivatives.
Sine, cosine and the unit circle
On the unit circle x² + y² = 1, the point at angle t has coordinates (cos t, sin t).
That single picture gives the range [−1, 1], the period 2π and the identity sin² t + cos² t = 1. The derivative of sine comes from the definition: expand sin(x + h) with the angle sum identity, then use the two special limits sin θ/θ → 1 and (cos θ − 1)/θ → 0 as θ → 0. The result is cos x, and a parallel argument gives −sin x for cosine. Radians are essential here.
Measured in degrees, the first special limit equals π/180 instead of 1, and that constant would multiply every trigonometric derivative.
Differentiating sine four times returns sine, a periodicity you can use to find high order derivatives quickly: the 103rd derivative of sin x, for instance, is −cos x because 103 leaves remainder 3 on division by 4.
Inverse functions and their slopes
Sine is not one to one on ℝ, so arcsin is defined by restricting sine to [−π/2, π/2], an interval on which it takes every value in [−1, 1] exactly once.
Arctangent restricts tangent to (−π/2, π/2) and accepts every real input. The inverse function rule says that if f′(c) ≠ 0 then (f⁻¹)′(x) = 1/f′(f⁻¹(x)). You do not need to memorise it: differentiate f(f⁻¹(x)) = x with the chain rule and solve. For arcsine this produces 1/cos(arcsin x), and a right triangle with opposite side x and hypotenuse 1 converts cos(arcsin x) into √(1 − x²).
The same method yields 1/(1 + x²) for arctangent, using sec² θ = 1 + tan² θ.
Watch the domains: arcsin is defined at ±1 but differentiable only on the open interval (−1, 1), since the slope formula has a zero denominator at the endpoints.
Exponentials, logarithms and logarithmic differentiation
For a positive base a, the difference quotient of aˣ factors as aˣ times (aʰ − 1)/h, so the derivative of aˣ is a fixed constant times aˣ.
Euler's number e is the base whose constant equals 1, giving (eˣ)′ = eˣ. Differentiating e^(ln x) = x shows that (ln x)′ = 1/x for x > 0, and writing aˣ = e^(x ln a) gives (aˣ)′ = (ln a) aˣ. Your notes may write log x for the natural logarithm; with no base shown, base e is intended. Logarithmic differentiation exploits ln(uv) = ln u + ln v and ln(uⁿ) = n ln u.
Taking ln of both sides turns a long product or quotient into a sum, and turns a variable power such as f(x)^g(x) into g(x) ln f(x), which the product rule handles. Before taking the logarithm, confirm the expression is positive, or work with ln|y|, whose derivative is still y′/y wherever y ≠ 0. Finally, multiply back by y: forgetting that step is the most common way a correct calculation ends with a wrong answer.
The generalised power rule (xⁿ)′ = n xⁿ⁻¹ for every real n follows the same route, by writing xⁿ = e^(n ln x).
What this chapter covers
- 01
Sine and cosine derivatives from the special limits
- 02
Tangent and secant via the quotient rule
- 03
Unit circle coordinates and reference angles
- 04
Inverse function rule and the arcsine derivative
- 05
Arctangent and its domain
- 06
Exponential and natural log derivatives
- 07
Logarithmic differentiation of products and variable powers
Worked example · free
Slope of x raised to ln x at x = e
- 1Since x > 0, y > 0, so take natural logarithms: ln y = (ln x)(ln x) = (ln x)².
- 2Differentiate implicitly using the chain rule on the right: y′/y = 2(ln x)(1/x) = 2 ln x / x.
- 1Multiply back by y: dy/dx = x^(ln x) · 2 ln x / x = 2 ln x · x^(ln x − 1).
- 2At x = e: ln e = 1, so y(e) = e¹ = e and dy/dx = e · 2(1)/e = 2.
Key terms
- Transcendental Function
- A transcendental function is a function such as sin x, arctan x, eˣ or ln x that is not algebraic. Its derivative must be obtained from limits, earlier differentiation rules, or an inverse relationship.
- Inverse Function Rule
- The inverse function rule is the formula (f⁻¹)′(x) = 1/f′(f⁻¹(x)), valid near a point where f is continuously differentiable with nonzero derivative. It follows from the chain rule applied to f(f⁻¹(x)) = x.
- Reference Angle
- A reference angle is the acute angle between the terminal side of an angle and the x axis. It fixes the absolute values of sine and cosine, while the quadrant decides their signs.
- Logarithmic Differentiation
- Logarithmic differentiation is a technique that takes ln of both sides of y = f(x), differentiates implicitly to find y′/y, then multiplies by y. It handles long products, quotients and variable exponents.
Trigonometric, Exponential and Log Derivatives FAQ
What is the derivative of arcsin x and where does it exist?
It equals 1/√(1 − x²). Arcsine accepts inputs in [−1, 1], but the slope formula only works on the open interval (−1, 1), because the denominator is zero at ±1 and the graph has vertical tangents at those two endpoints.
How do you differentiate a function raised to a function, like x to the power x?
Set y equal to the expression, take ln of both sides to get the exponent multiplied by ln of the base, differentiate implicitly with the product rule, then multiply through by y. Neither the ordinary power rule nor the exponential rule applies directly.
Why must calculus use radians for trigonometric derivatives?
The key limit sin θ/θ → 1 holds only when θ is in radians. With degrees the limit becomes π/180, and that stray constant multiplies every trigonometric derivative, breaking the clean pairing sin′ = cos and cos′ = −sin. Calculators set to degree mode cause the same error numerically.
Exam move
Rebuild the table rather than memorising it: on a blank page, derive sin′ from the special limits, tan′ from the quotient rule, arcsin′ from the inverse rule with a triangle, and (ln x)′ from e^(ln x) = x. Time yourself until the full set takes under ten minutes. Then practise logarithmic differentiation on three new functions, always writing the positivity check first and the multiply back by y step last.
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