a) For the function y= e^-3x, determine dy/dx, d^2/dx^2 and d^3y/dx^3.
b) From the pattern in part a, state the value for �((d^n)y)/(d(x^n)).
a)
dy/dx = (-3) * (e^-3x)
d^2/dx^2 = (-3) * (-3) * (e^-3x)
d^3y/dx^3 = (-3) * (-3) * (-3) * (e^-3x)
b)
Do you see a pattern above? What do you think the answer for this part should be based on that pattern?
Thank you very much! I appreciate the brisk response!
a) To determine dy/dx, the first derivative of y = e^(-3x), we need to apply the power rule for differentiation. The power rule states that if we have a function y = e^(kx), where k is a constant, then the derivative is dy/dx = ke^(kx).
In this case, our function is y = e^(-3x), so the constant is -3. Applying the power rule, we have dy/dx = (-3)e^(-3x) = -3e^(-3x).
To find d^2/dx^2, the second derivative of y, we differentiate dy/dx with respect to x. Following the power rule again, we have d^2y/dx^2 = -3(d/dx)(e^(-3x)).
Differentiating e^(-3x) with respect to x gives us (-3)e^(-3x) since the derivative of e^(-3x) is the same as the derivative of e^(kx) where k is -3.
Hence, d^2y/dx^2 = (-3)(-3)e^(-3x) = 9e^(-3x).
Finally, to find d^3y/dx^3, the third derivative of y, we differentiate d^2y/dx^2 with respect to x. Applying the power rule once more, we get d^3y/dx^3 = (d/dx)(9e^(-3x)).
Differentiating 9e^(-3x) with respect to x gives us 9(-3)e^(-3x) since the derivative of e^(-3x) is the same as the derivative of e^(kx) where k is -3.
Therefore, d^3y/dx^3 = (9)(-3)(-3)e^(-3x) = -27e^(-3x).
b) From the pattern we found in part a, we can state that the value of ((d^n)y)/(d(x^n)) for the function y = e^(-3x) is (-3)^n times e^(-3x). In other words, the value is (-3)^n multiplied by the original exponential function e^(-3x).