Chapter 2: Differentiation
Section 2.5: Implicit Differentiation
Obtain the derivative of y+x from x2+y2=9, the implicit representation of the circle.
Direct solution via the Context Panel
Control-drag the equation of the circle.
Context Panel: Differentiate≻Implicitly
(See explanation in next cell, below.)
After selecting "Implicitly" in the Context Panel, the dialog box shown in Figure 2.5.2(a) opens. Select y as the dependent variable. "Differentiate with respect to" x.
The result of the calculation matches the outcome found in Example 2.5.1, provided the letter y in the implicit derivative is replaced by 9−x2, the explicit expression for y+x.
Note that −x/y represents the derivative y+ and y−; simply replace y with y±x, as appropriate.
Figure 2.5.2(a) Dialog box for implicit derivative
Solution by task template
Calculus - Differential≻Derivatives≻Implicit Differentiation≻fx,y=gx,y
Enter an equation in two variables:
Dependent variable: Independent variable:
Make dependent variable explicit:
Differentiate with respect to independent variable:
Make independent variable implicit:
After entering the problem details, the Execute button returns the same derivative as the Context Panel.
The Stepwise Calculation illustrates the four basic steps of implicit differentiation. First, each letter y is replaced by yx; then, the expression is differentiated with respect to x. The term yx2 is differentiated with the Chain rule.
Clicking the Stepwise button launches the
tutor with the differentiation problem
embedded. The details of the differentiation can be explored via this tutor. Table 2.5.1(a) lists and annotates the steps as per the tutor.
Table 2.5.2(a) Annotated stepwise derivative as per the
In the third step, the derivative y′x is isolated.
In the final step, yx is restored to the letter y.
Interactively implemented stepwise solution
Control-drag (or type) the equation of the circle.
Context Panel: Evaluate at a Point≻y=yx
Context Panel: Differentiate≻With Respect To≻x
Solve≻Isolate Expression for≻diffyx,x
→evaluate at point
→differentiate w.r.t. x
→isolate for diff(y(x),x)
Solution via the implicitdiff command
implicitdiffx2+y2=9,y,x = −xy
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