How do you find the derivative of #y=e^x cos(x)# ?

Answer 1

#dy/dx=e^x (-sinx)+cosx(e^x)#

When two variables are multiplied in derivative We Have Formula, #(d(uv))/dx=u.((d(v))/dx)+ v.((d(u))/dx)# Question: To find the derivative of #y=e^xcos(x)# ?
Diffentiating both side by #x# We get : #dy/dx=(d(e^xcos(x)))/dx#
#dy/dx=e^x ((d(cosx))/dx)+cosx ((d(e^x ))/dx)#
Here, #(d(cosx))/dx=-sinx# for first derivative.
#dy/dx=e^x (-sinx)+cosx(e^x)#

which is the necessary resolution.

Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer 2

This kind of issue is one that makes use of the product rule.

According to the product rule:

#d/dx[f(x) * g(x)] = f'(x)g(x) + f(x)g'(x)#
So, we will let #f(x) = e^x#, and #g(x) = cos x#.
We know that the derivative of #e^x# is simply #e^x#, and that the derivative of #cos x# is equal to #-sin x#.
(if these identities look unfamiliar to you, I may recommend viewing videos from this page or this page, which explain the derivative rules for #e^x# and #cos x# more in-depth)
Therefore, #f'(x) = e^x#, and #g'(x) = -sin x#. We can then simply substitute into the product rule formula:
#d/dx[e^x cos x] = e^x * (-sin x) + e^x * cos x#
To make this equation a little prettier, we will factor the #e^x#:
#d/dx[e^x cos x] = e^x * (cos x - sin x)#
Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer 3

To find the derivative of ( y = e^x \cos(x) ), you can use the product rule of differentiation, which states that if ( u ) and ( v ) are differentiable functions of ( x ), then the derivative of their product is given by ( (uv)' = u'v + uv' ). Applying this rule:

[ \frac{d}{dx}(e^x \cos(x)) = e^x \cdot (-\sin(x)) + \cos(x) \cdot e^x ]

So, the derivative of ( y = e^x \cos(x) ) is:

[ y' = e^x \cdot (-\sin(x)) + \cos(x) \cdot e^x = e^x (-\sin(x) + \cos(x)) ]

Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

Not the question you need?

Drag image here or click to upload

Or press Ctrl + V to paste
Answer Background
HIX Tutor
Solve ANY homework problem with a smart AI
  • 98% accuracy study help
  • Covers math, physics, chemistry, biology, and more
  • Step-by-step, in-depth guides
  • Readily available 24/7