Differentiate from first principle:
i) y=x^2-3x
ii) y=5sinx
I will do one:
y'= Lim ((x+dx)^2-3(x+dx)-x^2+3x)/dx as dx>>zero
= Lim (x^2+2cdx+dx^2-3x-3dx-x^2+3x)/dx
=lim( 2x+dx-3) as dx>0
= 2x-3
To differentiate a function from first principles, we need to use the definition of the derivative. The derivative of a function represents the rate of change of the function with respect to its independent variable. It gives us the slope of the tangent line to the function at any given point.
i) Differentiating y = x^2 - 3x:
Step 1: Start with the function y = x^2 - 3x.
Step 2: Add a small change (h) to the x-value to get a new x-value: x + h.
Step 3: Calculate the corresponding change in the y-value by plugging in x + h into the original function: y + h = (x + h)^2 - 3(x + h).
Step 4: Find the difference in the y-values: (y + h) - y.
Step 5: Use the definition of the derivative: lim(h->0)((y + h) - y) / h.
Step 6: Simplify the expression: lim(h->0)((x^2 + 2xh + h^2 - 3x - 3h) - (x^2 - 3x)) / h.
Step 7: Expand and simplify the expression further: lim(h->0)(2xh + h^2 - 3h) / h.
Step 8: Cancel out the h factor in the numerator and denominator: lim(h->0)(2x + h - 3).
Step 9: Take the limit as h approaches 0: 2x - 3.
Therefore, the derivative of y = x^2 - 3x is dy/dx = 2x - 3.
ii) Differentiating y = 5sin(x):
Step 1: Start with the function y = 5sin(x).
Step 2: Add a small change (h) to the x-value to get a new x-value: x + h.
Step 3: Calculate the corresponding change in the y-value by plugging in x + h into the original function: y + h = 5sin(x + h).
Step 4: Find the difference in the y-values: (y + h) - y.
Step 5: Use the definition of the derivative: lim(h->0)((y + h) - y) / h.
Step 6: Simplify the expression: lim(h->0)(5sin(x + h) - 5sin(x)) / h.
Step 7: Apply the trigonometric identity: sin(a + b) = sin(a)cos(b) + cos(a)sin(b).
Step 8: Rewrite the expression using the trigonometric identity: lim(h->0)(5sin(x)cos(h) + 5cos(x)sin(h) - 5sin(x)) / h.
Step 9: Cancel out the h factor in the numerator and denominator: lim(h->0)(5sin(x)cos(h) + 5cos(x)sin(h) - 5sin(x)).
Step 10: Take the limit as h approaches 0: 5cos(x).
Therefore, the derivative of y = 5sin(x) is dy/dx = 5cos(x).