Master the algebra of derivatives in CBSE Class 11 Applied Mathematics. Learn sum, difference, product, and quotient rules with solved examples.
The derivative of the sum or difference of two differentiable functions is equal to the sum or difference of their individual derivatives. If f(x) and g(x) are differentiable, then the derivative of their combination is found by differentiating each term separately. This rule allows us to break down complex polynomial expressions into simpler parts.
Step 1: Apply the sum rule: d/dx(3x²) + d/dx(5x).
Step 2: Use the power rule: d/dx(3x²) = 6x and d/dx(5x) = 5.
Step 3: Combine the results: 6x + 5.
Answer: 6x + 5
When two functions are multiplied, the derivative is not simply the product of their derivatives. Instead, we use the product rule, which involves the sum of the first function times the derivative of the second and the second function times the derivative of the first. This is essential for differentiating expressions like x²·sin(x).
Step 1: Identify f(x) = x² and g(x) = eˣ.
Step 2: Find derivatives: f'(x) = 2x and g'(x) = eˣ.
Step 3: Apply formula: x²·eˣ + eˣ·2x.
Answer: eˣ(x² + 2x)
The quotient rule is used to differentiate a function that is a ratio of two other functions. It requires careful attention to the order of terms in the numerator and the squaring of the denominator. This rule is vital for rational functions where the denominator is a function of x.
Step 1: Identify f(x) = x, g(x) = eˣ.
Step 2: Derivatives: f'(x) = 1, g'(x) = eˣ.
Step 3: Apply formula: (eˣ·1 - x·eˣ) / (eˣ)².
Answer: (1 - x) / eˣ