Differentiation EF

Which of the following is NOT a correct differentiation rule?

• D_x [c] = 0

• D_x [xⁿ] = nx^n-1

• D_x [c f(x)] = c f'(x)

• D_x [f(x) + g(x)] = f'(x) + g'(x)

• D_x [f(x)/g(x)] = f'(x)/g'(x)

If f(x) = e^x - cos x, what is f'(x)?

• e^x - sin x

• e^x + sin x

• sin x - e^x

• • - sin x - e^x

• e^x sin x

Evaluate the following derivative: D_x [tan x²]

• sec² x²

• sec² 2x

• tan 2x

• 2x sec² x²

• 2x sec² 2x

What is the slope of the tangent line to the graph of y + sin y = x at the point (0, 0)?

• 1/2

• -1/2

• 0

• 1

• -1

If f(x) = 3x⁴ + 5x³, what is f'''(0)?

• 72

• 102

• 30

• 24

• 54

What is the equation of the normal line to the graph of f(x) = x² - x + 1 at its y-intercept?

At what point on the curve y = e^x + x does the tangent line have a slope of 2?

Suppose f, g, and h are differentiable functions such that f'(x) = g(x), g'(x) = -f(x), while h(x) = [f(x)]²+[g(x)]² for all real numbers x. If h(4) = 2025, find h(2025).