Question text Consider the functions $f(x)={3\,x^2+4\,x+5}$ and $g(x)={5\,x^2+3\,x}$. Find and simplify the derivative of ${4}f(x)-{5}g(x)$. [input] Your last answer was interpreted as follows: $$ -26\,x+1 $$The variables found in your answer were: $ \left[ x \right]$

Question

BlackTom AI · Accepted Answer

We start by identifying the given functions: f(x) = 3x^2 + 4x + 5 and g(x) = 5x^2 + 3x.

First, compute the derivatives of f and g separately:
f'(x) = d/dx[3x^2 + 4x + 5] = 6x + 4,
g'(x) = d/dx[5x^2 + 3x] = 10x + 3.

Next, form the linear combination 4f(x) − 5g(x) and differentiate using linearity: 
If h(x) = 4f(x) − 5g(x), then h'(x) = 4f'(x) − 5g'(x).

Substitute the computed derivatives:
h'(x) = 4(6x + 4) − 5(10x + 3).

Distribute the constants:
h'(x) = (24x + 16) − (50x + 15).

Now combine like terms:
24x − 50x = −26x, and 16 − 15 = 1. Therefore h'(x) = −26x + 1.

If you were asked to enter the derivative of 4f(x) − 5g(x) in simplified form, this is the result you would obtain: −26x + 1.

Note: There were no multiple-choice options provided in the prompt to evaluate; the reasoning above shows the full derivation step by step. Any discrepancy with a previously entered answer would come from a slips in applying the product/chain rules (not relevant here since f and g are polynomials with straightforward derivatives) or from arithmetic mistakes during distribution or combination of like terms.

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