Find the derivative of the following functions using first principles. a) f(x)= x^2 + 5^x

Answer 1

d/dx (x^2+5^x) = d/dx(x^2) + d/dx(5^x)
= 2 x + second half
for second half see:

https://ocw.mit.edu/courses/mathematics/18-01sc-single-variable-calculus-fall-2010/1.-differentiation/part-b-implicit-differentiation-and-inverse-functions/session-17-the-exponential-function-its-derivative-and-its-inverse/MIT18_01SCF10_Ses17c.pdf

d/dx 5^x = d/dx [ e^ln5]^x = d/dx e^(x ln5) = ln 5 e^(x ln 5)
= ln 5 * e^ln(5^x) = ln 5 * 5^x
so
2x + ( ln 5 * 5^x)