Most of the time, to take the derivative of a function given by a formula y = f(x), we can apply differentiation functions (refer to the table of derivative rules) along with the product, quotient, and chain rule. Sometimes though, it is not possible to solve and get an exact formula for y. For example, to find the derivative of y(x) given the function,
we must use implicit differentiation, since we cannot find y' explicitly as a function of only x. The best we can do is find a formula for y' as a function of both x and y. The resulting equation is called an implicit formula for y because we cannot solve for y directly.
To find y', we need to take the derivative of the entire equation with respect to x. Therefore, every time we encounter an expression involving functions of y, we need to use the chain rule since y itself is already a function of x. If we take the derivative of both sides with respect to x, we first apply the chain rule to y3 to get:
Note that the reason we use the d/dx notation is to emphasize that we are taking the derivative of the entire equation with respect to x, not y. This is why the derivative of has the y' factor, rather than simply being 3y2, which would be the case if we were differentiating with respect to y.
Similarly, we apply the chain rule and product rule with respect to x to sin(xy) to get:
On the right-hand side of the equation,
Plugging in the previous results, we get:
We couldn't solve for y in the original equation, but we can now solve for y'.
Below are a few more examples of implicit differentiation.
Use implicit differentiation to find the derivatives of the following equations.
1. Find the derivative with respect to x of :
2. Find the derivative with respect to x of :
First, apply the tangent function to the left and right sides of the equation:
Then, find the derivative with respect to x:
Using the trigonometric identity, and substituting , we can instead write the above equation as:
3. Find the derivative with respect to t of :