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Limits and Derivatives - Algebra of derivative of functions

Grade 11CBSE

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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The derivative of a constant times a function is equal to the constant times the derivative of the function: ddx[c⋅f(x)]=c⋅f′(x)\frac{d}{dx}[c \cdot f(x)] = c \cdot f'(x). Geometrically, multiplying a function by a constant c>1c > 1 stretches the graph vertically, which increases the slope of the tangent line at every point by that same factor cc.

Graph showing f(x) and 2f(x) illustrating vertical stretch and slope change.
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The Sum and Difference Rules state that the derivative of a sum or difference of two functions is the sum or difference of their individual derivatives: ddx[f(x)±g(x)]=f′(x)±g′(x)\frac{d}{dx}[f(x) \pm g(x)] = f'(x) \pm g'(x). This allows for term-by-term differentiation of polynomials.

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The Product Rule (Leibniz Rule) handles the derivative of two functions multiplied together: ddx[u(x)v(x)]=u′(x)v(x)+u(x)v′(x)\frac{d}{dx}[u(x)v(x)] = u'(x)v(x) + u(x)v'(x). It is not simply the product of the derivatives.

Visual representation of product rule using area increments du and dv.
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The Quotient Rule is used for functions in the form of a fraction: ddx(uv)=vu′−uv′v2\frac{d}{dx}\left(\frac{u}{v}\right) = \frac{v u' - u v'}{v^2}. Always ensure the denominator v(x)≠0v(x) \neq 0.

Flowchart for applying the quotient rule.

📐Formulae

f′(x)=lim⁡h→0f(x+h)−f(x)hf'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}

ddx(xn)=nxn−1\frac{d}{dx}(x^n) = nx^{n-1}

ddx(c)=0\frac{d}{dx}(c) = 0 (where cc is a constant)

ddx[f(x)±g(x)]=f′(x)±g′(x)\frac{d}{dx}[f(x) \pm g(x)] = f'(x) \pm g'(x)

ddx[u(x)⋅v(x)]=u′(x)v(x)+u(x)v′(x)\frac{d}{dx}[u(x) \cdot v(x)] = u'(x)v(x) + u(x)v'(x)

ddx[u(x)v(x)]=u′(x)v(x)−u(x)v′(x)[v(x)]2\frac{d}{dx}\left[\frac{u(x)}{v(x)}\right] = \frac{u'(x)v(x) - u(x)v'(x)}{[v(x)]^2}

ddx(sin⁡x)=cos⁡x\frac{d}{dx}(\sin x) = \cos x

ddx(cos⁡x)=−sin⁡x\frac{d}{dx}(\cos x) = -\sin x

💡Examples

Problem 1:

Find the derivative of the function f(x)=x3+4x2−5f(x) = x^3 + 4x^2 - 5 with respect to xx.

Solution:

Step 1: Identify that the function is a sum of terms. We apply the Sum Rule and Constant Multiple Rule. Step 2: Differentiate each term individually. ddx(x3)=3x3−1=3x2\frac{d}{dx}(x^3) = 3x^{3-1} = 3x^2 ddx(4x2)=4⋅(2x2−1)=8x\frac{d}{dx}(4x^2) = 4 \cdot (2x^{2-1}) = 8x ddx(−5)=0\frac{d}{dx}(-5) = 0 (since the derivative of a constant is zero). Step 3: Combine the results. f′(x)=3x2+8xf'(x) = 3x^2 + 8x.

Explanation:

This problem uses the basic Power Rule and the linearity of the derivative (sum rule). Each power of xx is reduced by 1, and the coefficient is multiplied by the original exponent.

Problem 2:

Differentiate y=x2cos⁡xy = x^2 \cos x using the Product Rule.

Solution:

Step 1: Identify the two functions. Let u(x)=x2u(x) = x^2 and v(x)=cos⁡xv(x) = \cos x. Step 2: Find the derivatives of uu and vv. u′(x)=2xu'(x) = 2x v′(x)=−sin⁡xv'(x) = -\sin x Step 3: Apply the Product Rule formula: dydx=u′v+uv′\frac{dy}{dx} = u'v + uv'. dydx=(2x)(cos⁡x)+(x2)(−sin⁡x)\frac{dy}{dx} = (2x)(\cos x) + (x^2)(-\sin x) Step 4: Simplify the expression. dydx=2xcos⁡x−x2sin⁡x\frac{dy}{dx} = 2x \cos x - x^2 \sin x.

Explanation:

To differentiate a product of two different types of functions (algebraic and trigonometric), the Product Rule is essential. We calculate the derivative of each part separately and then combine them according to the formula.

Problem 3:

Find the derivative of the function f(x)=x+1x−1f(x) = \frac{x+1}{x-1} with respect to xx where x≠1x \neq 1.

Graph of (x+1)/(x-1) showing the vertical asymptote at x=1.

Solution:

Let u=x+1u = x+1 and v=x−1v = x-1. Then u′=1u' = 1 and v′=1v' = 1. Using the Quotient Rule: f′(x)=(x−1)(1)−(x+1)(1)(x−1)2f'(x) = \frac{(x-1)(1) - (x+1)(1)}{(x-1)^2} f′(x)=x−1−x−1(x−1)2f'(x) = \frac{x - 1 - x - 1}{(x-1)^2} f′(x)=−2(x−1)2f'(x) = \frac{-2}{(x-1)^2}

Explanation:

To differentiate a rational function, we identify the numerator uu and denominator vv, find their respective derivatives, and apply the Quotient Rule formula.

Problem 4:

Differentiate y=5sin⁡x+x4y = 5 \sin x + x^4 with respect to xx.

Graph of the function 5sin(x) + x^4.

Solution:

Apply the Sum Rule and the Constant Multiple Rule: dydx=ddx(5sin⁡x)+ddx(x4)\frac{dy}{dx} = \frac{d}{dx}(5 \sin x) + \frac{d}{dx}(x^4) dydx=5ddx(sin⁡x)+4x4−1\frac{dy}{dx} = 5 \frac{d}{dx}(\sin x) + 4x^{4-1} dydx=5cos⁡x+4x3\frac{dy}{dx} = 5 \cos x + 4x^3

Explanation:

The derivative of a sum is the sum of the derivatives. We use the known derivative of sin⁡x\sin x and the power rule for xnx^n.