Question

attempt 1: 10 attempts remaining. given ( f(x) = (7x^3 - 3x^2 + 8x - 5)^{10} ), find ( f(x) ). ( f(x) = ) input box submit answer next item

Explanation:

Step1: Identify the outer and inner functions

Let \( u = 7x^3 - 3x^2 + 8x - 5 \) (inner function) and \( y = u^{10} \) (outer function). We will use the chain rule, which states that \( \frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx} \).

Step2: Differentiate the outer function with respect to u

For \( y = u^{10} \), using the power rule \( \frac{d}{du}(u^n)=nu^{n - 1} \), we have \( \frac{dy}{du}=10u^{9} \).

Step3: Differentiate the inner function with respect to x

For \( u = 7x^3 - 3x^2 + 8x - 5 \), using the power rule for each term:

• The derivative of \( 7x^3 \) is \( 7\times3x^{2}=21x^{2} \)
• The derivative of \( - 3x^2 \) is \( -3\times2x=-6x \)
• The derivative of \( 8x \) is \( 8 \)
• The derivative of \( - 5 \) (a constant) is \( 0 \)

So, \( \frac{du}{dx}=21x^{2}-6x + 8 \).

Step4: Apply the chain rule

Substitute \( \frac{dy}{du} \) and \( \frac{du}{dx} \) into the chain rule formula \( \frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx} \). Replace \( u \) with \( 7x^3 - 3x^2 + 8x - 5 \):

\( f^{\prime}(x)=10(7x^{3}-3x^{2}+8x - 5)^{9}\cdot(21x^{2}-6x + 8) \)

Answer:

\( 10(21x^{2}-6x + 8)(7x^{3}-3x^{2}+8x - 5)^{9} \)