Find the derivative of the following functions using first principles. a) f(x)= x^2 + 5^x
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)
To find the derivative of a function using first principles, we need to use the definition of the derivative. The definition of the derivative of a function f(x) at a point x=a is:
f'(a) = lim(h->0) [f(a+h) - f(a)] / h
Let's apply this definition to the function f(x) = x^2 + 5^x.
a) f(x) = x^2 + 5^x
Let's find the derivative of f(x) using first principles.
Step 1: Substitute the function into the definition of the derivative:
f'(a) = lim(h->0) [f(a+h) - f(a)] / h
f'(x) = lim(h->0) [(x+h)^2 + 5^(x+h) - (x^2 + 5^x)] / h
Step 2: Expand the terms to simplify the expression:
f'(x) = lim(h->0) [(x^2 + 2xh + h^2) + 5^x * 5^h - (x^2 + 5^x)] / h
Step 3: Combine like terms:
f'(x) = lim(h->0) [2xh + h^2 + 5^x * 5^h - 5^x] / h
Step 4: Factor out the common terms:
f'(x) = lim(h->0) h(2x + h) / h + lim(h->0) 5^x (5^h - 1) / h
Step 5: Cancel out the 'h' term:
f'(x) = lim(h->0) 2x + h + lim(h->0) 5^x (5^h - 1) / h
Step 6: Take the limit as h approaches 0:
f'(x) = 2x + lim(h->0) h + lim(h->0) 5^x (5^h - 1) / h
Step 7: Evaluate the limits:
f'(x) = 2x + 0 + 5^x * lim(h->0) (5^h - 1) / h
Step 8: Simplify the expression:
f'(x) = 2x + 5^x * lim(h->0) (5^h - 1) / h
Step 9: Observe that the expression inside the limit is now in a different form. This expression is a familiar derivative:
f'(x) = 2x + 5^x * ln(5)
Therefore, the derivative of f(x) = x^2 + 5^x using first principles is f'(x) = 2x + 5^x * ln(5).