If y = e^x cosx, prove that dy/dx sqrt(2) e^x . cos(x+ pie/4)
The simple dy/dx = e^x(-sinx) + e^xcosx
= e^x(cosx - sinx)
RS = √2(e^x)(cos(x+π/4)
= √2(e^x)(cosxcosπ/4 - sinxsinπ/4)
= √2(e^x)(cosx(1/√2 - sinx(1/√2)
= e^x (cosx - sinx)
= the dy/dx from above
To prove that the derivative of y = e^x cos(x) is equal to √2 e^x cos(x + π/4), we can use the product rule of differentiation. The product rule states that if we have two functions u(x) and v(x), then the derivative of their product is given by:
(d/dx) [u(x) * v(x)] = u'(x) * v(x) + u(x) * v'(x)
Let's start by finding the derivative of y = e^x cos(x). We will use the product rule with u(x) = e^x and v(x) = cos(x).
Step 1: Find u'(x)
The derivative of u(x) = e^x is given by u'(x) = d/dx (e^x).
The derivative of e^x with respect to x is simply e^x.
Therefore, u'(x) = e^x.
Step 2: Find v'(x)
The derivative of v(x) = cos(x) is given by v'(x) = d/dx (cos(x)).
The derivative of the cosine function is given by -sin(x).
Therefore, v'(x) = -sin(x).
Step 3: Apply the product rule
Using the product rule, the derivative of y = e^x cos(x) is given by:
dy/dx = u'(x) * v(x) + u(x) * v'(x)
= (e^x)(cos(x)) + (e^x)(-sin(x))
= e^x * (cos(x) - sin(x))
Now, we need to manipulate the expression e^x * (cos(x) - sin(x)) to make it match √2 e^x cos(x + π/4).
Step 4: Rewrite the expression
We know that √2 = 1.4142 (approximately), so we can rewrite √2 e^x as 1.4142 e^x.
Also, using the trigonometric identity cos(x + π/4) = cos(x)cos(π/4) - sin(x)sin(π/4), we can rewrite cos(x) - sin(x) as cos(x + π/4).
Therefore, e^x * (cos(x) - sin(x)) can be rewritten as 1.4142 e^x cos(x + π/4).
Step 5: Final result
Comparing our rewritten expression with the derivative we found earlier:
dy/dx = e^x * (cos(x) - sin(x))
We see that dy/dx = 1.4142 e^x cos(x + π/4), which matches the given expression. Hence, we have proved that dy/dx = √2 e^x cos(x + π/4).