3.14: Derivatives of Inverse Trig Functions (2024)

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    In this section we explore the relationship between the derivative of a function and the derivative of its inverse. For functions whose derivatives we already know, we can use this relationship to find derivatives of inverses without having to use the limit definition of the derivative. In particular, we will apply the formula for derivatives of inverse functions to trigonometric functions. This formula may also be used to extend the power rule to rational exponents.

    The Derivative of an Inverse Function

    Note: The Inverse Function Theorem is an "extra" for our course, but can be very useful. There are other methods to derive (prove) the derivatives of the inverse Trigonmetric functions. Be sure to see the Table of Derivatives of Inverse Trigonometric Functions.

    We begin by considering a function and its inverse. If \(f(x)\) is both invertible and differentiable, it seems reasonable that the inverse of \(f(x)\) is also differentiable. Figure shows the relationship between a function \(f(x)\) and its inverse \(f^{−1}(x)\). Look at the point \((a,f^{−1}(a))\) on the graph of \(f^{−1}(x)\) having a tangent line with a slope of

    \[(f−1)′(a)=\dfrac{p}{q}.\]

    This point corresponds to a point \((f^{−1}(a),a)\) on the graph of \(f(x)\) having a tangent line with a slope of

    \[f′(f^{−1}(a))=\dfrac{q}{p}.\]

    Thus, if \(f^{−1}(x)\) is differentiable at \(a\), then it must be the case that

    \((f^{−1})′(a)=\dfrac{1}{f′(f^{−1}(a))}\).

    3.14: Derivatives of Inverse Trig Functions (2)

    Figure \(\PageIndex{1}\):The tangent lines of a function and its inverse are related; so, too, are the derivatives of these functions.

    We may also derive the formula for the derivative of the inverse by first recalling that \(x=f(f^{−1}(x))\). Then by differentiating both sides of this equation (using the chain rule on the right), we obtain

    \(1=f′(f^{−1}(x))(f^{−1})′(x))\).

    Solving for \((f^{−1})′(x)\), we obtain

    \((f^{−1})′(x)=\dfrac{1}{f′(f^{−1}(x))}\).

    We summarize this result in the following theorem.

    Inverse Function Theorem

    Let \(f(x)\) be a function that is both invertible and differentiable. Let \(y=f^{−1}(x)\) be the inverse of \(f(x)\). For all \(x\) satisfying \(f′(f^{−1}(x))≠0\),

    \[\dfrac{dy}{dx}=\dfrac{d}{dx}(f^{−1}(x))=(f^{−1})′(x)=\dfrac{1}{f′(f^{−1}(x))}.\]

    Alternatively, if \(y=g(x)\) is the inverse of \(f(x)\), then

    \[g(x)=\dfrac{1}{f′(g(x))}.\]

    Example \(\PageIndex{1}\): Applying the Inverse Function Theorem

    Use the inverse function theorem to find the derivative of \(g(x)=\dfrac{x+2}{x}\). Compare the resulting derivative to that obtained by differentiating the function directly.

    Solution

    The inverse of \(g(x)=\dfrac{x+2}{x}\) is \(f(x)=\dfrac{2}{x−1}\).

    Since \[g′(x)=\dfrac{1}{f′(g(x))},\]

    begin by finding \(f′(x)\). Thus,

    \(f′(x)=\dfrac{−2}{(x−1)^2}\) and \(f′(g(x))=\dfrac{−2}{(g(x)−1)^2}=\dfrac{−2}{(\dfrac{x+2}{x}−1)^2}=−\dfrac{x^2}{2}\).

    Finally,

    \(g′(x)=\dfrac{1}{f′(g(x))}=−\dfrac{2}{x^2}\).

    We can verify that this is the correct derivative by applying the quotient rule to \(g(x)\) to obtain

    \(g′(x)=−\dfrac{2}{x^2}\).

    3.14: Derivatives of Inverse Trig Functions (3) Exercise \(\PageIndex{1}\)

    Use the inverse function theorem to find the derivative of \(g(x)=\dfrac{1}{x+2}\). Compare the result obtained by differentiating \(g(x)\) directly.

    Hint

    Use the preceding example as a guide.

    Answer

    \(g′(x)=−\dfrac{1}{(x+2)^2}\)

    Example \(\PageIndex{2}\): Applying the Inverse Function Theorem

    Use the inverse function theorem to find the derivative of \(g(x)=\sqrt[3]{x}\).

    Solution

    The function \(g(x)=\sqrt[3]{x}\) is the inverse of the function \(f(x)=x^3\). Since \(g′(x)=\dfrac{1}{f′(g(x))}\), begin by finding \(f′(x)\). Thus,

    \[f′(x)=3x^3\]

    and

    \[f′(g(x))=3(\sqrt[3]{x})^2=3x^{2/3}\]

    Finally,

    \[g′(x)=\dfrac{1}{3x^{2/3}}=\dfrac{1}{3}x^{−2/3}.\]

    3.14: Derivatives of Inverse Trig Functions (4) Exercise \(\PageIndex{2}\)

    Find the derivative of \(g(x)=\sqrt[5]{x}\) by applying the inverse function theorem.

    Hint

    \(g(x)\) is the inverse of \(f(x)=x^5\).

    Answer

    \(g(x)=\dfrac{1}{5}x^{−4/5}\)

    From the previous example, we see that we can use the inverse function theorem to extend the power rule to exponents of the form \(\dfrac{1}{n}\), where \(n\) is a positive integer. This extension will ultimately allow us to differentiate \(x^q\), where \(q\) is any rational number.

    Extending the Power Rule to Rational Exponents

    The power rule may be extended to rational exponents. That is, if \(n\) is a positive integer, then

    \[\dfrac{d}{dx}(x^{1/n})=\dfrac{1}{n} x^{(1/n)−1}.\]

    Also, if \(n\) is a positive integer and \(m\) is an arbitrary integer, then

    \(\dfrac{d}{dx}(x^{m/n})=\dfrac{m}{n}x^{(m/n)−1}\).

    Proof

    The function \(g(x)=x^{1/n}\) is the inverse of the function \(f(x)=x^n\). Since \(g′(x)=\dfrac{1}{f′(g(x))}\), begin by finding \(f′(x)\). Thus,

    \(f′(x)=nx^{n−1}\) and \(f′(g(x))=n(x^{1/n})^{n−1}=nx^{(n−1)/n}\).

    Finally,

    \(g′(x)=\dfrac{1}{nx^{(n−1)/n}}=\dfrac{1}{n}x^{(1−n)/n}=\dfrac{1}{n}x^{(1/n)−1}\).

    To differentiate \(x^{m/n}\) we must rewrite it as \((x^{1/n})^m\) and apply the chain rule. Thus,

    \[\dfrac{d}{dx}(x^{m/n})=\dfrac{d}{dx}((x^{1/n})^m)=m(x^{1/n})^{m−1}⋅\dfrac{1}{n}x^{(1/n)−1}=\dfrac{m}{n}x^{(m/n)−1}.\]

    Example \(\PageIndex{3}\): Applying the Power Rule to a Rational Power

    Find the equation of the line tangent to the graph of \(y=x^{2/3}\) at \(x=8\).

    Solution

    First find \(\dfrac{dy}{dx}\) and evaluate it at \(x=8\). Since

    \(\dfrac{dy}{dx}=\dfrac{2}{3}x^{−1/3}\) and \(\dfrac{dy}{dx}∣_{x=8}=\dfrac{1}{3}\)

    the slope of the tangent line to the graph at \(x=8\) is \(\dfrac{1}{3}\).

    Substituting \(x=8\) into the original function, we obtain \(y=4\). Thus, the tangent line passes through the point \((8,4)\). Substituting into the point-slope formula for a line, we obtain the tangent line

    \(y=\dfrac{1}{3}x+\dfrac{4}{3}\).

    3.14: Derivatives of Inverse Trig Functions (5) Exercise \(\PageIndex{3}\)

    Find the derivative of \(s(t)=\sqrt{2t+1}\).

    Hint

    Use the chain rule.

    Answer

    \(s′(t)=(2t+1)^{−1/2}\)

    Derivatives of Inverse Trigonometric Functions

    We now turn our attention to finding derivatives of inverse trigonometric functions. These derivatives will prove invaluable in the study of integration later in this text. The derivatives of inverse trigonometric functions are quite surprising in that their derivatives are actually algebraic functions. Previously, derivatives of algebraic functions have proven to be algebraic functions and derivatives of trigonometric functions have been shown to be trigonometric functions. Here, for the first time, we see that the derivative of a function need not be of the same type as the original function.

    Example \(\PageIndex{4}\): Derivative of the Inverse Sine Function

    Use the inverse function theorem to find the derivative of \(g(x)=\sin ^{−1}x\).

    Solution

    Since for \(x\) in the interval \([−\dfrac{π}{2},\dfrac{π}{2}],f(x)=\sin x\) is the inverse of \(g(x)=sin^{−1}x\), begin by finding \(f′(x)\). Since

    \(f′(x)=\cos x\) and \(f′(g(x))=\cos (\sin ^{−1}x)=\sqrt{1−x^2}\),

    we see that

    \(g′(x)=\dfrac{d}{dx}(\sin ^{−1}x)=\dfrac{1}{f′(g(x))}=\dfrac{1}{\sqrt{1−x^2}}\).

    Analysis

    To see that \(\cos (\sin^{−1}x)=\sqrt{1−x^2}\), consider the following argument. Set \(\sin ^{−1}x=θ\). In this case, \(\sin θ=x\) where \(−\dfrac{π}{2}≤θ≤\dfrac{π}{2}\). We begin by considering the case where \(0<θ<\dfrac{π}{2}\). Since \(θ\) is an acute angle, we may construct a right triangle having acute angle \(θ\), a hypotenuse of length \(1\) and the side opposite angle \(θ\) having length \(x\). From the Pythagorean theorem, the side adjacent to angle \(θ\) has length \(\sqrt{1−x^2}\). This triangle is shown in Figure. Using the triangle, we see that \(\cos (\sin ^{−1}x)=\cos θ=\sqrt{1−x^2}\).

    3.14: Derivatives of Inverse Trig Functions (6)

    Figure \(\PageIndex{2}\): Using a right triangle having acute angle \(θ\), a hypotenuse of length \(1\), and the side opposite angle \(θ\) having length \(x\), we can see that \(\cos (\sin ^{−1}x)=\cos θ=\sqrt{1−x^2}\).

    In the case where \(−\dfrac{π}{2}<θ<0\), we make the observation that \(0<−θ<\dfrac{π}{2}\) and hence

    \(\cos (\sin ^{−1}x)=\cos θ=\cos (−θ)=\sqrt{1−x^2}\).

    Now if \(θ=\dfrac{π}{2}\) or \(θ=−\dfrac{π}{2},x=1\) or \(x=−1\), and since in either case \(\cos θ=0\) and \(\sqrt{1−x^2}=0\), we have

    \(\cos (\sin ^{−1}x)=\cos θ=\sqrt{1−x^2}\).

    Consequently, in all cases, \(\cos (\sin ^{−1}x)=\sqrt{1−x^2}\).

    Example \(\PageIndex{5}\): Applying the Chain Rule to the Inverse Sine Function

    Apply the chain rule to the formula derived in Example to find the derivative of \(h(x)=sin^{−1}(g(x))\) and use this result to find the derivative of \(h(x)=sin^{−1}(2x^3).\)

    Solution

    Applying the chain rule to \(h(x)=\sin ^{−1}(g(x))\), we have

    \(h′(x)=\dfrac{1}{\sqrt{1−(g(x))^2}}g′(x)\).

    Now let \(g(x)=2x^3,\) so \(g′(x)=6x.\) Substituting into the previous result, we obtain

    \(h′(x)=\dfrac{1}{\sqrt{1−4x^6}}⋅6x=\dfrac{6x}{\sqrt{1−4x^6}}\)

    3.14: Derivatives of Inverse Trig Functions (7) Exercise \(\PageIndex{4}\)

    Use the inverse function theorem to find the "derive" the derivative of \(g(x)=\tan ^{−1}x\).

    Hint

    The inverse of \(g(x)\) is \(f(x)=\tan x\). Use Example \(\PageIndex{5}\) as a guide.

    Answer

    \(g′(x)=\dfrac{1}{1+x^2}\)

    The derivatives of the remaining inverse trigonometric functions may also be found by using the inverse function theorem. These formulas are provided in the following theorem.

    Table of Derivatives of Inverse Trigonometric Functions

    \(\dfrac{d}{dx}\sin ^{−1}x=\dfrac{1}{\sqrt{1−(x)^2}}\)

    \(\dfrac{d}{dx}\cos ^{−1}x=\dfrac{−1}{\sqrt{1−(x)^2}}\)

    \(\dfrac{d}{dx}\tan ^{−1}x=\dfrac{1}{1+(x)^2}\)

    \(\dfrac{d}{dx}\cot ^{−1}x=\dfrac{−1}{1+(x)^2}\)

    \(\dfrac{d}{dx}\sec ^{−1}x=\dfrac{1}{|x|\sqrt{(x)^2−1}}\)

    \(\dfrac{d}{dx}\csc ^{−1}x=\dfrac{−1}{|x|\sqrt{(x)^2−1}}\)

    Example \(\PageIndex{6}\): Applying Differentiation Formulas to an Inverse Tangent Function

    Find the derivative of \(f(x)=\tan ^{−1}(x^2).\)

    Solution

    \(Let g(x)=x^2\), so \(g′(x)=2x.\) Substituting into Equation, we obtain

    \(f′(x)=\dfrac{1}{1+(x^2)^2}⋅(2x).\)

    Simplifying, we have

    \(f′(x)=\dfrac{2x}{1+x^4}\).

    Example \(\PageIndex{7}\): Applying Differentiation Formulas to an Inverse Sine Function

    Find the derivative of \(h(x)=x^2\sin ^{−1}x.\)

    Solution

    By applying the product rule, we have

    \(h′(x)=2x\sin ^{−1}x+\dfrac{1}{\sqrt{1−x^2}}⋅x2\)

    3.14: Derivatives of Inverse Trig Functions (8) Exercise \(\PageIndex{5}\)

    Find the derivative of \(h(x)=\cos ^{−1}(3x−1).\)

    Hint

    Use Equation. with \(g(x)=3x−1\)

    Answer

    \(h′(x)=\dfrac{−3}{\sqrt{6x−9x^2}}\)

    Example \(\PageIndex{8}\): Applying the Inverse Tangent Function

    The position of a particle at time \(t\) is given by \(s(t)=\tan ^{−1}(\dfrac{1}{t})\) for \(t≥\dfrac{1}{2}\). Find the velocity of the particle at time \( t=1\).

    Solution

    Begin by differentiating \(s(t)\) in order to find \(v(t)\).Thus,

    \(v(t)=s′(t)=\dfrac{1}{1+(\dfrac{1}{t})^2}⋅\dfrac{−1}{t^2}\).

    Simplifying, we have

    \(v(t)=−\dfrac{1}{t^2+1}\).

    Thus, \(v(1)=−\dfrac{1}{2}.\)

    3.14: Derivatives of Inverse Trig Functions (9) Exercise \(\PageIndex{6}\)

    Find the equation of the line tangent to the graph of \(f(x)=\sin ^{−1}x\) at \(x=0.\)

    Hint

    \(f′(0)\) is the slope of the tangent line.

    Answer

    \(y=x\)

    Key Concepts

    • The inverse function theorem allows us to compute derivatives of inverse functions without using the limit definition of the derivative.
    • We can use the inverse function theorem to develop differentiation formulas for the inverse trigonometric functions.

    Key Equations

    • Inverse function theorem

    \((f−1)′(x)=\dfrac{1}{f′(f^{−1}(x))}\) whenever \(f′(f^{−1}(x))≠0\) and \(f(x)\) is differentiable.

    • Power rule with rational exponents

    \(\dfrac{d}{dx}(x^{m/n})=\dfrac{m}{n}x^{(m/n)−1}.\)

    • Derivative of inverse sine function

    \(\dfrac{d}{dx}\sin^{−1}x=\dfrac{1}{\sqrt{1−(x)^2}}\)

    • Derivative of inverse cosine function

    \(\dfrac{d}{dx}\cos^{−1}x=\dfrac{−1}{\sqrt{1−(x)^2}}\)

    Derivative of inverse tangent function

    \(\dfrac{d}{dx}\tan^{−1}x=\dfrac{1}{1+(x)^2}\)

    Derivative of inverse cotangent function

    \(\dfrac{d}{dx}\cot^{−1}x=\dfrac{−1}{1+(x)^2}\)

    Derivative of inverse secant function

    \(\dfrac{d}{dx}\sec^{−1}x=\dfrac{1}{\sqrt{|x|(x)^2−1}}\)

    Derivative of inverse cosecant function

    \(\dfrac{d}{dx}\csc^{−1}x=\dfrac{−1}{|x|\sqrt{(x)^2−1}}\)

    Contributors

    • Gilbert Strang (MIT) and Edwin “Jed” Herman (Harvey Mudd) with many contributing authors. This content by OpenStax is licensedwith a CC-BY-SA-NC4.0license. Download for free at http://cnx.org.

    I'm well-versed in the topic of calculus and derivatives. Now, let's delve into the concepts discussed in the provided article.

    The article primarily focuses on the relationship between the derivative of a function and the derivative of its inverse. The key concepts mentioned include the Inverse Function Theorem, which allows us to find derivatives of inverse functions without resorting to the limit definition of the derivative.

    Here's a breakdown of the main points:

    1. Inverse Function Theorem:

      • If (f(x)) is both invertible and differentiable, and (y=f^{−1}(x)) is the inverse of (f(x)), then ((f^{−1})′(x)=\dfrac{1}{f′(f^{−1}(x))}).
    2. Applying the Inverse Function Theorem:

      • Examples are provided to illustrate the application of the Inverse Function Theorem in finding derivatives. For instance, the derivative of (g(x)=\dfrac{x+2}{x}) is found using the Inverse Function Theorem.
    3. Extending the Power Rule:

      • The power rule is extended to rational exponents, allowing differentiation of expressions like (x^{m/n}).
    4. Derivatives of Inverse Trigonometric Functions:

      • Formulas for finding derivatives of inverse trigonometric functions are given using the Inverse Function Theorem. Examples include the derivatives of (\sin ^{−1}x), (\cos ^{−1}x), (\tan ^{−1}x), (\cot ^{−1}x), (\sec ^{−1}x), and (\csc ^{−1}x).
    5. Application to Particle Motion:

      • An example involves finding the velocity of a particle at a given time, where the position is described by an inverse trigonometric function.

    The article provides clear explanations, examples, and formulas to enhance understanding of these calculus concepts. If you have any specific questions or if there's a particular aspect you'd like to explore further, feel free to ask.

    3.14: Derivatives of Inverse Trig Functions (2024)
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