Question

Can we write derivative of 5(2x-3)^-3 directly 5*-3(2x-3)^-3-1 *coefficient of x, instead of using chain rule.In what type of question this is applicable and in which question not?Please explain.

Answer 1

Don't say "instead of using the chain rule", since the result you stated comes from the chain rule

I given y = 5(2x-3)^-3 I would write the derivative as
dy/dx = -15(2x-3)^-3 * 2 = -30(2x-3)^-4

The detailed steps would only be done when you first learn how the process works. Once you have mastered the procedure, you can state the answer directly in usually 2 steps

e.g y = 3(3x^3 - 7)^7
dy/dx = 21(3x^3 - 7)^6 * 9x^2
= 189x^2(3x^3-7)^6

Answer 2

Why can't we do this for questions like f(X)=15/(2x+1)^1/3

Answer 3

We can!

dy/dx = (1/3)(15)(2x+1)^(1/3 -1) * 2
= 10(2x+1)^(-2/3) , done in 2 steps!

Answer 4

[AI-generated answer — Explain Bot]

To find the derivative of the function 5(2x-3)^-3, we cannot directly apply the formula you mentioned, which appears to be a misapplication of the power rule. Instead, we need to use the chain rule, which is applicable to any situation where we have a composite function.

The power rule, in general, states that to take the derivative of a function of the form f(x) = cx^n, where c is a constant, we can multiply the coefficient by the exponent and then decrease the exponent by 1. However, in your case, we have a composite function, where the base function is (2x-3) raised to the power -3.

To apply the chain rule, we need to split the function into two parts: the outer function and the inner function. The outer function is f(x) = 5x^-3, and the inner function is g(x) = 2x-3. The composite function is formed by nesting the inner function within the outer function: f(g(x)) = 5(2x-3)^-3.

Using the chain rule, the derivative of the composite function is given by the product of the derivative of the outer function with respect to the inner function (df/dg) and the derivative of the inner function with respect to x (dg/dx). Applying the chain rule, we have:

df/dx = df/dg * dg/dx

For the outer function f(g(x)) = 5x^-3, we can apply the power rule directly, finding df/dg = -15x^-4.

For the inner function g(x) = 2x-3, the derivative dg/dx is simply 2.

Now, we can apply the chain rule:

df/dx = df/dg * dg/dx
= -15x^-4 * 2
= -30x^-4

Therefore, using the chain rule, the derivative of 5(2x-3)^-3 is -30x^-4.

In summary, the direct approach you mentioned would lead to an incorrect answer. The chain rule is applicable whenever we have a composite function, where we have one function nested inside another. We need to differentiate both the outer and inner functions separately and then multiply their derivatives to obtain the derivative of the composite function. The direct approach you mentioned may be applicable in simple cases where only the power rule is sufficient, without any nested or composite functions.