i have a math question, can you please help?

Question

dy/dx(1)= x^5+3e^xsin(x)+e^3
they want me to find the derivative of that andplug in one, but i keep gtting stuck in the e and sinx section. can someone help me ?

Answer 1

Hi Dear;

If f(x) = x^5 + 3e^xsin(x) + e^3

Part A;
y = x^5
Differentiating x^5 with respect to x:
Apply the "power rule" to x^5 (the derivative of xn=nxn-1) giving 5x^4
y' = 5x^4

Part B;
if u = 3e^x ; u'= 3e^x
{as u know if f(x) = e^x so f'(x) = e^x }
& v = sin(x) ; v' = cos(x)
f'(x) = u'v +v'u
f'(x) = 3e^xsin(x) + 3e^x cos(x)


Part C;
y = e^3
e^3does not contain any reference to the variable x. So the derivative
of this term with respect to x is zero (0) (it is a constant with respect to x).
so y' = 0

--Total RESULT:
f(x) = x^5+3e^xsin(x)+e^3
f'(x) = 5x^4 + 3e^xsin(x) + 3e^x cos(x)
{ here you can Factor ' 3e^x' }
f'(x) = 5x^4 + 3e^x(sin(x) + cos(x))
now fill ' 1 ' in to the final result;
f'(1) = 5(1)^4 + 3e^(1)((sin(1) + cos(1))

It is The Correct answer.
Good Luck Dear.

Answer 2

I believe the question is to evaluate the derivative at x = 1, i.e. d/dx (y(x))@x=1 The derivative is 5x^4 + 3e^xsin(x) + 3e^xcos(x). At x=1, this is 5 + 0 + 3*e = 5 + 3*e. EDIT: 5 + 3^e*sin(1) + 3*e*cos(1) = 5 + 3*.841*e + 3*.54*e = 16.268

Answer 3

derivative of l of x^5 = 5x^4

derivative of e^xsin(x)
using y =uv formula = e^x(d/dx(sin x) + sin x d/dx(e^x)
= e^x(cos x) + e^x(sinx) = e^x(cos x+ sin x)
derivative of e^3 = 0

so dy/dx = 5x^4+e^x(sin x+ cos x)
if at 1 then we mean
at x =1 the value is 5*1^4+e^1(sin 1 + cos 1) = 5+ e(sin 1+ cos 1)