Lesson Taking derivative of a function, which is defined implicitly
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<H2>Taking derivative of a function, which is defined implicitly<H2> <H3>Problem 1</H3><pre>Let y = sin (t), x = ln (t). It defines y as a function of x, y = y(x), implicitly. d^2 y Find ------- when x = {{{pi}}}. dx^2 </pre> <B>Solution</B> <pre> If x = ln(t), then t = {{{e^x}}}, (1) where "e" is the base of natural logarithms. Therefore, in this problem, after making substitution (1), we have y = sin(e^x), (2) i.e. function y is expressed as the composition of function sine and exponent. So, we apply the formula for the derivative of a composite function and find first derivative of y with respect to x {{{(dy)/dx)}}} = {{{cos(e^x)*((d(e^x))/(dx))}}} = {{{cos(e^x)*e^x}}}. (3) Then we find second derivative as the derivative of (3) d^2 y ------- (x) = {{{-sin(e^x)*e^x*e^x}}} + {{{cos(e^x)*e^x}}} = {{{-sin(e^x)*e^(2x)}}} + {{{cos(e^x)*e^x}}}. dx^2 Now we substitute x = {{{pi}}} to get d^2 y ------- {{{(pi)}}} = {{{-sin(e^pi)*e^(2pi)}}} + {{{cos(e^pi)*e^pi}}}. dx^2 To get the value, use in calculations approximate values e = 2.71828, pi = 3.14159. d^2 y ------- {{{(pi)}}} = {{{-sin(2.71828^3.14159)*2.71828^(2*3.14159)}}} + {{{cos(2.71828^3.14159)*2.71828^3.14159}}} = 479.18428 (rounded). dx^2 </pre> My other lessons on Calculus word problems at this site are - <A HREF=https://www.algebra.com/algebra/homework/word/misc/A-ladder-foot-slides-on-the-ground-txt.lesson>A ladder foot slides on the ground</A> - <A HREF=https://www.algebra.com/algebra/homework/word/misc/Finding-rate-of-change-of-some-processes.lesson>Finding rate of change of some processes</A> - <A HREF=https://www.algebra.com/algebra/homework/word/misc/Find-the-derivative-of-a-function-defined-by-complicated-expression.lesson>Find the derivative of a function defined by complicated expression</A> - <A HREF=https://www.algebra.com/algebra/homework/word/misc/Find-the-derivative-of-a-function-satisfying-given-functional-equation.lesson>Find the derivative for a function satisfying given functional equation</A> - <A HREF=https://www.algebra.com/algebra/homework/word/misc/09-add-Find-the-Range-of-f%28x%29-=-%285cos%28x%29%29-div-%28x-%2B-1%29%29cos%28x%29.lesson>Find the range of f(x) = (5*cos(x))/(x + 1)), x >=0</A> - <A HREF=https://www.algebra.com/algebra/homework/word/misc/A-tricky-Calculus-problem-on-derivative-and-anti-derivative.lesson>A tricky Calculus problem on derivative and anti-derivative</A> - <A HREF=https://www.algebra.com/algebra/homework/word/misc/Couple-of-non-standard-Calculus-problems.lesson>Couple of non-standard Calculus problems</A> - <A HREF=https://www.algebra.com/algebra/homework/word/misc/Finding-the-minimum-of-a-function-defined-on-a-curve-in-the-coordinate-plane.lesson>Finding the minimum of a function defined on a curve in a coordinate plane</A> - <A HREF=https://www.algebra.com/algebra/homework/word/misc/Tricky-solution-to-find-the-maximum-of-a-function-defined-by-a-complicated-expression.lesson>Tricky solution to find the maximum of a function defined by a complicated expression</A> - <A HREF=https://www.algebra.com/algebra/homework/word/misc/Maximize-the-area-of-a-trapezoid.lesson>Maximize the area of a trapezoid </A> - <A HREF=https://www.algebra.com/algebra/homework/word/misc/Maximmize-the-volume-of-an-open-box.lesson>Maximize the volume of an open box</A> - <A HREF=https://www.algebra.com/algebra/homework/word/misc/Minimize-surface-area-of-a-rectangular-prizm-with-the-given-volume.lesson>Minimize surface area of a rectangular box with the given volume</A> - <A HREF=https://www.algebra.com/algebra/homework/word/misc/Minimize-the-cost-of-an-aquarium-with-the-given-volume.lesson>Minimize the cost of an aquarium with the given volume</A> - <A HREF=https://www.algebra.com/algebra/homework/word/misc/Minimize-surface-area-of-a-conic-paper-cup-with-the-given-volume.lesson>Minimize surface area of a conical paper cup with the given volume</A> - <A HREF=https://www.algebra.com/algebra/homework/word/misc/Find-the-volume-of-a-solid-body-obtained-by-rotation-the-area-about-an-axis.lesson>Find the volume of a solid obtained by rotation of some plane shape about an axis</A> - <A HREF=https://www.algebra.com/algebra/homework/word/misc/Finding-the-volume-of-a-solid-body-mentally.lesson>Finding the volume of a solid body mentally</A> - <A HREF=https://www.algebra.com/algebra/homework/word/misc/OVERVIEW-of-my-lessons-on-Calculus-word-problems.lesson>OVERVIEW of my lessons on Calculus word problems</A> Use this file/link <A HREF=https://www.algebra.com/algebra/homework/complex/ALGEBRA-II-YOUR-ONLINE-TEXTBOOK.lesson>ALGEBRA-II - YOUR ONLINE TEXTBOOK</A> to navigate over all topics and lessons of the online textbook ALGEBRA-II.
