High School

Use the chain rule to find the derivative of [tex]3e^{10x^6 - 9x^7}[/tex].

Use [tex]e^x \cdot x[/tex] for [tex]e^x[/tex].

Answer :

Final answer:

To find the derivative of the given function using the chain rule, we need to identify the outer and inner functions, find their respective derivatives, and then multiply them together.

Explanation:

Derivative of 3e^(10x-9x^2)

To find the derivative of the given function, we can use the chain rule. The chain rule states that if we have a composition of two functions, f(g(x)), the derivative is given by f'(g(x)) * g'(x).

Let's apply the chain rule:

  1. Identify the outer function f(u) and the inner function u.
  2. Find the derivative of the outer function: f'(u).
  3. Find the derivative of the inner function: u'.
  4. Multiply the derivatives obtained in steps 2 and 3.

In this case, the outer function is 3e^u and the inner function is 10x-9x^2. Let's find the derivatives:

  • The derivative of 3e^u with respect to u is 3e^u.
  • The derivative of 10x-9x^2 with respect to x is 10-18x.

Multiplying these derivatives, we get (3e^(10x-9x^2))(10-18x).

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