Question

Find the derivative of the function.
F(x) = (x4 + 7x2 − 9)5
F'(x) =

Answer 1

To find the derivative of the function \( F(x) = (x^4 + 7x^2 - 9)^5 \), we can use the chain rule. The chain rule states that the derivative of a composite function \( F(g(x)) \) is given by \( F'(g(x)) \cdot g'(x) \).

Here, let \( u = x^4 + 7x^2 - 9 \). Then, we can write:

\[
F(x) = u^5
\]

Now we'll differentiate \( F \) with respect to \( u \) and then multiply by the derivative of \( u \) with respect to \( x \):

1. Differentiate \( F(u) = u^5 \):
\[
\frac{dF}{du} = 5u^4
\]

2. Now, we need to find \( \frac{du}{dx} \):
\[
u = x^4 + 7x^2 - 9
\]
\[
\frac{du}{dx} = 4x^3 + 14x
\]

3. Now apply the chain rule:
\[
F'(x) = \frac{dF}{du} \cdot \frac{du}{dx} = 5u^4 \cdot (4x^3 + 14x)
\]

4. Substitute \( u \) back in:
\[
F'(x) = 5(x^4 + 7x^2 - 9)^4 \cdot (4x^3 + 14x)
\]

Thus, the derivative \( F'(x) \) is:

\[
F'(x) = 5(x^4 + 7x^2 - 9)^4 (4x^3 + 14x)
\]