Final answer:
To find the differential dy when x = 1 and dx = 0.4 in the equation y = tan(5x + 7), we can use the chain rule of differentiation. The chain rule states that if y = f(g(x)), then dy/dx = f'(g(x)) * g'(x). Substituting the values into the derivative formula, we can find that dy/dx is approximately equal to 7.675.
Step-by-step explanation:
The given equation is y = tan(5x + 7). We need to find the value of dy when x = 1 and dx = 0.4.
To do this, we can use the chain rule of differentiation. The chain rule states that if y = f(g(x)), then dy/dx = f'(g(x)) * g'(x).
In this case, f(u) = tan(u) and g(x) = 5x + 7. Taking the derivative of f(u) gives f'(u) = sec^2(u), and the derivative of g(x) is g'(x) = 5.
Substituting the values into the chain rule, we get dy/dx = sec^2(5x + 7) * 5.
When x = 1 and dx = 0.4, we can plug these values into the derivative formula to find the value of dy.
dy/dx = sec^2(5*1 + 7) * 5 = sec^2(12) * 5.
Using a calculator, sec(12) ≈ 1.239 and sec^2(12) ≈ 1.535. So, dy/dx ≈ 1.535 * 5 = 7.675.
Therefore, when x = 1 and dx = 0.4, the differential dy is approximately equal to 7.675.