1.
Find f '(−4), if f(x) = (5x^2 + 6x)(3x^2 + 7). Round your answer to the
nearest integer. Use the hyphen symbol, -, for negative values.
f(x) = 15x^4 + 35x^2 + 18x^3 + 42x
f'(x) = 15*4 x^3 + 2*35 x + 3*18 x^2 + 42
f'(x) = 60x^3 + 70x + 58x^2 + 42
f'( - 4) = 60( -4)^3 + 70 (- 4) + 58( -4)^2 + 42 = - 3150
Answer: - 3150
2. Find f'(x) for f(x) = −7x^2 + 4x − 10.
f'(x) = - 2*7x + 4 = -14x + 4
Answer: −14x + 4
3.If f and g are differentiable functions for all real values of x such
that f(1) = 4, g(1) = 3, f '(3) = −5, f '(1) = −4, g '(1) = −3, g '(3) =
2, then find h '(1) if h(x) = f(x) g(x).
h (x) = f(x) g(x) => h '(x) =. chain rule => f '(x) g(x) + f(x) g '(x)
h'(1) = f ' (1) g(1) + f(1) g '(1) = - 4 * 3 + 4 * ( - 3) = - 12 - 12 = - 24
Answer: - 24
4.Find the coefficient of the squared term in the simplified form for
the second derivative, f "(x) for f(x) = (x^3 + 2x + 3)(3x^3 − 6x^2 − 8x +
1). Use the hyphen symbol, -, for negative values.
f(x) = 3x^6 - 6x^3 - 8x^4 + x^3 + 6x^5 - 12x^3 - 16x^2 + 2x + 9x^3 - 18x^2 - 24x + 3 = 3x^6 + 6x^5 - 8x^4 - 8x^3 - 34x^2 + 26x + 3
f '(x) = 18x^5 + 30x^4 - 32x^3 -24x^2 - 68x + 26
f ''(x) = 90x^4 + 120x^3 - 96x^2 - 48x - 68
So the coefficient of the squared term is - 96.
You can tell that without all the calculus if you realize that the squared term comes from the term with the power 4 (because when you find the second derivative the power decreases two units). And that term is - 8x^4
And the second derivative of -8x^4 is -8*4*3 x^2 = -96x^2, where you see the coefficient is -96.
Answer: - 96
