Solve by using first principle
A) f(x) = 3x² + x
B) f(x) = tanx
A) To find the derivative of f(x) = 3x² + x using the first principle, we will start by evaluating the difference quotient.
The difference quotient is given by:
[f(x + h) - f(x)] / h
= [(3(x + h)² + (x + h)) - (3x² + x)] / h
= [(3(x² + 2hx + h²) + x + h) - (3x² + x)] / h
= [(3x² + 6hx + 3h² + x + h) - (3x² + x)] / h
= (3h² + 6hx + h) / h
Cancel out the common factor of h:
= 3h + 6x + 1
Now, taking the limit as h approaches 0:
lim(h→0) (3h + 6x + 1) = 6x + 1
Therefore, the derivative of f(x) = 3x² + x is 6x + 1.
B) To find the derivative of f(x) = tanx using the first principle, we will again evaluate the difference quotient.
The difference quotient is given by:
[f(x + h) - f(x)] / h
= [tan(x + h) - tanx] / h
Since tan(x + h) cannot be simplified easily, we will use the trigonometric identity:
tan(x + h) - tanx = [tan(x) + tan(h)] / [1 - tan(x)tan(h)]
Therefore, the difference quotient becomes:
= [tan(x) + tan(h)] / [h * (1 - tan(x)tan(h))]
Now, taking the limit as h approaches 0:
lim(h→0) [tan(x) + tan(h)] / [h * (1 - tan(x)tan(h))]
Using the properties of limits and trigonometric identities, we can simplify this expression further:
= sec²(x) / (1 - tan²(x))
= sec²(x) / sec²(x) (since 1 - tan²(x) = sec²(x))
= 1
Therefore, the derivative of f(x) = tanx is 1.
A) To find the derivative of f(x) = 3x² + x using the first principle, we need to evaluate the following limit:
f'(x) = lim(h->0) [f(x + h) - f(x)] / h
Let's begin by calculating f(x + h):
f(x + h) = 3(x + h)² + (x + h)
= 3(x² + 2hx + h²) + x + h
= 3x² + 6hx + 3h² + x + h
Now, we substitute f(x) and f(x + h) into the formula:
f'(x) = lim(h->0) [3x² + 6hx + 3h² + x + h - (3x² + x)] / h
= lim(h->0) [6hx + 3h² + h] / h
Expanding further:
f'(x) = lim(h->0) 6hx/h + 3h²/h + h/h
= lim(h->0) 6x + 3h + 1
Since h approaches zero, we can eliminate the h-terms:
f'(x) = 6x + 1
Therefore, the derivative of f(x) = 3x² + x using the first principle is f'(x) = 6x + 1.
B) To find the derivative of f(x) = tan(x) using the first principle, we need to evaluate the following limit:
f'(x) = lim(h->0) [f(x + h) - f(x)] / h
Let's begin by calculating f(x + h):
f(x + h) = tan(x + h)
Now, we substitute f(x) and f(x + h) into the formula:
f'(x) = lim(h->0) [tan(x + h) - tan(x)] / h
By using the trigonometric identity for tan(x + h) - tan(x), we have:
f'(x) = lim(h->0) [tan(x) + tan(h) - tan(x)] / h
= lim(h->0) [tan(h)] / h
This limit does not have a finite value, so we cannot find the derivative of f(x) = tan(x) using the first principle. The derivative of tan(x) can be found using other differentiation rules or formulas, such as the chain rule.