Can anyone plz post the procedure for finding the derivative of :-
e^ax
by ab inition method?
Derivative by the ab initio (Lat. "from the beginning") method is to find it from first principles, i.e.
df(x)/dx
=Lim h->0 (f(x+h)-f(x))/h
for f(x)=e^ax, we have
Lim h->0 (e^a(x+h) - e^ax)/h
=Lim h->0 (e^ax*e^ah - e^ax)/h
=Lim h->0 e^ax(e^ah-1)/h
=e^ax Lim h->0 (e^ah-1)/h
Try to evaluate the limit, which should turn out to be a.
Can you plz evaluate the limit.
Lim h->0 (e^ah-1)/h
I'm having problems with it. Thanks
To find the derivative of the function e^ax using the "ab initio" or first principles method, you need to follow a few steps. Here's how you can do it:
Step 1: Start with the definition of the derivative. The derivative of a function f(x) is the limit of the difference quotient as the change in x approaches zero:
f'(x) = lim(h -> 0) [(f(x + h) - f(x))/h]
Step 2: Substitute the given function f(x) = e^ax into the difference quotient:
f'(x) = lim(h -> 0) [(e^a(x + h) - e^ax) / h]
Step 3: Apply the properties of exponents. Using the rule e^(a + b) = e^a * e^b, we can rewrite the numerator as:
f'(x) = lim(h -> 0) [e^ax * (e^ah - 1) / h]
Step 4: Factor out e^ax from the numerator:
f'(x) = e^ax * lim(h -> 0) [(e^ah - 1) / h]
Step 5: Now, we need to evaluate the limit as h approaches zero. This is equivalent to finding the derivative of e^ah with respect to h and then evaluating it at h = 0.
Step 6: Differentiate e^ah with respect to h. The derivative of e^ah is ae^ah:
d/dh (e^ah) = ae^ah
Step 7: Evaluate the derivative at h = 0:
d/dh (e^ah) | h=0 = a * e^a0 = a
Step 8: Substitute back into the original expression:
f'(x) = e^ax * a
Therefore, the derivative of e^ax with respect to x is a * e^ax.
Note: The "ab initio" method, also known as the first principles or limit definition method, involves taking the limit and evaluating it step by step. There are other methods, such as using the chain rule or the rules of exponentiation, which can provide a shortcut for calculating derivatives.