Implicit Differentiation
56.1 Implicit Functions
An implicit function is a function that is not explicitly defined in terms of one variable as a function of the other (e.g., $y = f(x)$). Instead, it's often expressed as a relation between variables, such as $x^2 + y^2 = 25$ (a circle) or $e^{xy} = x + y$. For these functions, we cannot easily isolate $y$ to differentiate using standard rules.
In such cases, $y$ is considered an implicit function of $x$. When differentiating with respect to $x$, any term involving $y$ must be treated using the chain rule, as if $y$ were $f(x)$.
MathCrave Option Solver (Implicit Functions)
Ask about implicit functions, e.g., 'What is an implicit function? Give an example.'
56.2 Differentiating Implicitly
Implicit differentiation is a technique used to find the derivative of an implicit function. The core idea is to differentiate both sides of the equation with respect to the independent variable (usually $x$), treating $y$ as a function of $x$ and applying the chain rule whenever a term involving $y$ is differentiated.
For example, if you differentiate $y^2$ with respect to $x$, it becomes $2y \frac{dy}{dx}$.
MathCrave Option Solver (Differentiating Implicitly)
Enter an implicit equation to differentiate, e.g., 'Differentiate $x^2 + y^2 = 25$ with respect to x.'
56.3 Products & Quotients (Implicit)
When applying implicit differentiation, you often encounter terms that require the product rule or quotient rule, especially when $x$ and $y$ are multiplied or divided within a term. Remember to apply the chain rule correctly to any term involving $y$ during these operations.
For instance, differentiating $xy$ with respect to $x$ requires the product rule: $\frac{d}{dx}(xy) = y \frac{d}{dx}(x) + x \frac{d}{dx}(y) = y(1) + x \frac{dy}{dx} = y + x \frac{dy}{dx}$.
MathCrave Option Solver (Products & Quotients Implicit)
Differentiate an equation with product/quotient terms, e.g., 'Find $\frac{dy}{dx}$ for $x^2y^3 = \sin(y)$.'
56.4 Advanced Implicit Differentiation
Advanced implicit differentiation problems might involve more complex functions (e.g., logarithmic, exponential, inverse trigonometric), or require finding higher-order derivatives (like $\frac{d^2y}{dx^2}$). The same principles apply: differentiate term by term, treating $y$ as a function of $x$, and use the chain rule. For higher-order derivatives, you'll differentiate the $\frac{dy}{dx}$ expression found in the first differentiation step, again applying implicit differentiation where needed.
MathCrave Option Solver (Advanced Implicit Differentiation)
Enter a complex implicit differentiation problem, e.g., 'Find $\frac{dy}{dx}$ for $e^x + e^y = e^{x+y}$.'