Find dy/dx tan(xy)=x
The problem presents a mathematical expression that relates a function of x and y, specifically the tangent of their product tan(xy), to x itself. The task is to find the derivative of y with respect to x, which is represented by the notation dy/dx. This requires the use of implicit differentiation, a technique used to differentiate functions when y is defined implicitly by an equation in terms of x, rather than being given as an explicit function y=f(x).
$tan \left(\right. x y \left.\right) = x$
Answer
Solution:
Step:1
Apply differentiation to both sides of the given equation, $\tan(xy) = x$.
Step:2
Differentiate the left-hand side.
Step:2.1
Invoke the chain rule for differentiation, which is $\frac{d}{dx}[f(g(x))] = f'(g(x))g'(x)$, where $f(x) = \tan(x)$ and $g(x) = xy$.
Step:2.1.1
Set $u = xy$ and differentiate $\tan(u)$ with respect to $u$ and $u$ with respect to $x$: $\frac{d}{du}[\tan(u)]\frac{d}{dx}[xy]$.
Step:2.1.2
Compute the derivative of $\tan(u)$ with respect to $u$: $(\sec^2(u))\frac{d}{dx}[xy]$.
Step:2.1.3
Substitute $u$ back with $xy$: $(\sec^2(xy))\frac{d}{dx}[xy]$.
Step:2.2
Apply the product rule for differentiation, $\frac{d}{dx}[f(x)g(x)] = f(x)g'(x) + g(x)f'(x)$, where $f(x) = x$ and $g(x) = y$: $(\sec^2(xy))(x\frac{d}{dx}[y] + y\frac{d}{dx}[x])$.
Step:2.3
Express $\frac{d}{dx}[y]$ as $y'$: $(\sec^2(xy))(xy' + y)$.
Step:2.4
Utilize the power rule, $\frac{d}{dx}[x^n] = nx^{n-1}$, where $n = 1$: $(\sec^2(xy))(xy' + y \cdot 1)$.
Step:2.5
Multiply $y$ by 1: $(\sec^2(xy))(xy' + y)$.
Step:2.6
Simplify the expression.
Step:2.6.1
Apply the distributive property: $(\sec^2(xy))xy' + (\sec^2(xy))y$.
Step:2.6.2
Reorder terms: $x(\sec^2(xy))y' + y(\sec^2(xy))$.
Step:3
Differentiate the right-hand side using the power rule: $1$.
Step:4
Combine the differentiated left and right sides: $x(\sec^2(xy))y' + y(\sec^2(xy)) = 1$.
Step:5
Isolate $y'$ (dy/dx).
Step:5.1
Simplify the left side.
Step:5.1.1
Factor out common terms: $xy(\sec^2(xy)) + y(\sec^2(xy)) = 1$.
Step:5.2
Subtract $y(\sec^2(xy))$ from both sides: $xy(\sec^2(xy)) = 1 - y(\sec^2(xy))$.
Step:5.3
Divide by $x(\sec^2(xy))$ to isolate $y'$.
Step:5.3.1
Divide each term: $\frac{xy(\sec^2(xy))}{x(\sec^2(xy))} = \frac{1}{x(\sec^2(xy))} - \frac{y(\sec^2(xy))}{x(\sec^2(xy))}$.
Step:5.3.2
Simplify the left side.
Step:5.3.2.1
Cancel out common factors: $\frac{y}{1} = \frac{1}{x(\sec^2(xy))} - \frac{y}{x}$.
Step:5.3.2.2
Simplify the right side: $y = \frac{1}{x(\sec^2(xy))} - \frac{y}{x}$.
Step:6
Replace $y$ with $\frac{dy}{dx}$: $\frac{dy}{dx} = \frac{1}{x(\sec^2(xy))} - \frac{y}{x}$.
Knowledge Notes:
Chain Rule: A fundamental rule in calculus used when differentiating a composite function. If $f$ and $g$ are functions, then the derivative of the composite function $f(g(x))$ is $f'(g(x))g'(x)$.
Product Rule: This rule is used when differentiating the product of two functions. If $u(x)$ and $v(x)$ are functions of $x$, then the derivative of their product is given by $u'(x)v(x) + u(x)v'(x)$.
Power Rule: A basic differentiation rule that states if $f(x) = x^n$, then $f'(x) = nx^{n-1}$.
Secant Function: The secant function, denoted as $\sec(x)$, is the reciprocal of the cosine function. It is defined as $\sec(x) = \frac{1}{\cos(x)}$.
Implicit Differentiation: This technique is used when it is difficult or impossible to solve an equation for one variable in terms of another. It involves differentiating both sides of the equation with respect to $x$ and then solving for $\frac{dy}{dx}$.
Simplification: The process of reducing an expression to its simplest form by performing algebraic operations like factoring, expanding, canceling common factors, and combining like terms.
