Find the derivative of each
1) y = e^x(Sin²x)
2) y = (x²+1)e^-4
3) y = x²(e^Sinx)
2) 2*x*e^(-4)
3) 2*x*e^(sin(x))+x^2*cos(x)*e^(sin(x))
1) e^(x)*sin(x)^2+2*e^(x)*sin(x)*cos(x)
To find the derivatives of the given functions, we can use the chain rule and product rule of differentiation. Here's how you can find the derivative for each function:
1) y = e^x(Sin²x):
To find the derivative of this function, we can use the product rule. The product rule states that if we have a function f(x) = g(x) * h(x), then its derivative is given by f'(x) = g'(x) * h(x) + g(x) * h'(x).
Let's break down the given function:
g(x) = e^x
h(x) = sin²x
Now, let's find the derivatives of g(x) and h(x):
g'(x) = d/dx(e^x) = e^x
h'(x) = d/dx(sin²x) = 2sin(x)cos(x) [using the chain rule]
Now, applying the product rule:
f'(x) = g'(x) * h(x) + g(x) * h'(x)
= e^x * sin²x + e^x * 2sin(x)cos(x)
Therefore, the derivative of y = e^x(Sin²x) is f'(x) = e^x * sin²x + e^x * 2sin(x)cos(x).
2) y = (x²+1)e^-4:
To find the derivative of this function, we can use the product rule again.
Let's break down the given function:
g(x) = x² + 1
h(x) = e^-4
Now, let's find the derivatives of g(x) and h(x):
g'(x) = d/dx(x² + 1) = 2x
h'(x) = d/dx(e^-4) = -4e^-4 [using the chain rule]
Now, applying the product rule:
f'(x) = g'(x) * h(x) + g(x) * h'(x)
= (2x) * e^-4 + (x² + 1) * (-4e^-4)
Therefore, the derivative of y = (x²+1)e^-4 is f'(x) = (2x) * e^-4 + (x² + 1) * (-4e^-4).
3) y = x²(e^sinx):
To find the derivative of this function, we can use the product rule.
Let's break down the given function:
g(x) = x²
h(x) = e^sinx
Now, let's find the derivatives of g(x) and h(x):
g'(x) = d/dx(x²) = 2x
h'(x) = d/dx(e^sinx) = e^sinx * cosx [using the chain rule]
Now, applying the product rule:
f'(x) = g'(x) * h(x) + g(x) * h'(x)
= (2x) * e^sinx + x² * e^sinx * cosx
Therefore, the derivative of y = x²(e^sinx) is f'(x) = (2x) * e^sinx + x² * e^sinx * cosx.
Note: Please double-check the solutions as some of these calculations can be more complex, and errors can occur.