Find an algebraic expression for f'
f(x)= 2e^x+ pi
f(x)=e^5-4e^x
f(x)=e^x+x^e+e
f(x)= e^pi +pi^x+ x^pi
I didn't understand these at all
2 e^x
-4 e^x because e^5 is a constant
e^x + e x^(e-1) because e is constant
pi^x ln pi + pi x^(pi-1) because e^pi is constant
To find the algebraic expression for the derivative (f') of the given functions, we can use the rules of differentiation. The derivative of a function measures its rate of change or slope at each point.
1. f(x) = 2e^x + pi:
To find f', we can take the derivative of each term separately. The derivative of 2e^x is simply 2e^x since the derivative of e^x is e^x. The derivative of pi is 0 since it is a constant. Thus, f'(x) = 2e^x + 0 = 2e^x.
2. f(x) = e^5 - 4e^x:
Similar to the previous example, the derivative of e^5 is 0 because it is a constant term. The derivative of -4e^x is simply -4e^x since the derivative of e^x is e^x. Therefore, f'(x) = 0 - 4e^x = -4e^x.
3. f(x) = e^x + x^e + e:
Here, we need to use the chain rule to differentiate x^e. The derivative of e^x is e^x, and when e is treated as a constant, the derivative of x^e is e * x^(e-1). Therefore, f'(x) = e^x + e * x^(e-1) + 0 = e^x + ex^(e-1).
4. f(x) = e^pi + pi^x + x^pi:
Taking the derivative of each term separately, the derivative of e^pi is 0 since it is a constant. The derivative of pi^x can be calculated using the chain rule, which results in (ln(pi)) * (pi^x). Finally, the derivative of x^pi can be found using the power rule, yielding pi * x^(pi-1). Thus, f'(x) = 0 + (ln(pi)) * (pi^x) + pi * x^(pi-1).
These are the algebraic expressions for the derivatives (f') of the given functions.