Question

Use implicit differentiation to find da/dt if a^4 - t^4 = 6a²t.

a) da/dt = (2a³ - 2t³)/(3a² - t²)
b) da/dt = (a³ - t³)/(2a² - t²)
c) da/dt = (3a² - 3t²)/(4a - t)
d) da/dt = (4a³ - 4t³)/(5a² - t²)

Answer 1

To find da/dt using implicit differentiation, differentiate each term of the equation with respect to t. Substitute the derivatives back into the equation and simplify to find the expression for da/dt.

Step-by-step explanation:

To find da/dt using implicit differentiation, we start by taking the derivative of the equation a^4 - t^4 = 6a^2t with respect to t. We treat a as a function of t and differentiate each term using the chain rule. The derivative of a^4 is 4a^3 * da/dt and the derivative of t^4 is 4t^3. The derivative of 6a^2t is 2a^2 * da/dt + 6a^3.

Substituting these derivatives back into the original equation, we get 4a^3 * da/dt - 4t^3 = 2a^2 * da/dt + 6a^3. Rearranging the terms, we have 2a^2 * da/dt - 4a^3 = 4t^3 - 6a^3. Simplifying further, we get da/dt = (4t^3 - 6a^3)/(2a^2 - 4a^3), which can be further reduced to da/dt = (2t^3 - 3a^3)/(a^2 - 2a^3).