Question

Find derivative of
a f(x)=x⁴ ln (x³+7)
b f(x)=4 x³+{3 x² {x³+7}

Answer 1

a)
`\( f'(x) = 4x^3 \ln(x^3 + 7) + (3x^3)/(x^3 + 7) \)`

b)
`\( f'(x) = 12x^2 + 18x^3 \)`

Step-by-step explanation:

a) To find the derivative of
`\( f(x) = x^4 \ln(x^3 + 7) \)`, we use the product rule, which states that if
`\( f(x) = g(x) \cdot h(x) \), then \( f'(x) = g'(x) \cdot h(x) + g(x) \cdot h'(x) \).`

In this case, let
`\( g(x) = x^4 \) and \( h(x) = \ln(x^3 + 7) \).` Applying the product rule:

`\[ f'(x) = 4x^3 \ln(x^3 + 7) + (3x^3)/(x^3 + 7) \]`

b) To find the derivative of
`\( f(x) = 4x^3 + 3x^2(x^3 + 7) \),` we use the sum rule and the product rule.

Apply the sum rule to
`\( 4x^3 \) and \( 3x^2(x^3 + 7) \),` and then apply the product rule to
`\( 3x^2(x^3 + 7) \):`

`\[ f'(x) = 12x^2 + 18x^3 \]`

These results give the derivatives of the given functions.