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Q3:
Differentiate the functions with respect to $x$. $\sin(ax + b)$
Differentiate the functions with respect to $x$. $\sin(ax + b)$
Solution :
Step 1: Formal Representation of the Function
Let the given function be represented as $y = f(x)$. Thus, we define:
$y = \sin(ax + b)$
[Where $a$ and $b$ are constants and $x$ is the independent variable].
Step 2: Identification of the Differentiation Rule
To differentiate this function, we must apply the Chain Rule [Calculus Theorem: If $y = f(g(x))$, then $\frac{dy}{dx} = f'(g(x)) \cdot g'(x)$].
Here, the composite function consists of an outer function $f(u) = \sin(u)$ and an inner function $u = g(x) = ax + b$.
Step 3: Application of Derivatives
Differentiating both sides with respect to $x$:
$\frac{dy}{dx} = \frac{d}{dx} [\sin(ax + b)]$
Applying the standard derivative formula $\frac{d}{dx}(\sin u) = \cos u \cdot \frac{du}{dx}$:
$\frac{dy}{dx} = \cos(ax + b) \cdot \frac{d}{dx}(ax + b)$
Step 4: Evaluating the Internal Derivative
Now, compute the derivative of the inner function $(ax + b)$ using the Sum Rule and Constant Multiple Rule of derivatives:
$\frac{d}{dx}(ax + b) = \frac{d}{dx}(ax) + \frac{d}{dx}(b)$
Since $a$ is a constant, $\frac{d}{dx}(ax) = a \cdot \frac{d}{dx}(x) = a(1) = a$.
Since $b$ is a constant, $\frac{d}{dx}(b) = 0$ [Derivative of a constant is zero].
Therefore, $\frac{d}{dx}(ax + b) = a$.
Step 5: Final Substitution and Simplification
Substitute the result from Step 4 back into the expression from Step 3:
$\frac{dy}{dx} = \cos(ax + b) \cdot (a)$
$\frac{dy}{dx} = a \cos(ax + b)$
Final Solution: $\frac{dy}{dx} = a \cos(ax + b)$
More Questions from Class 12 Mathematics Continuity and Differentiability EXERCISE 5.2
- Q1: Differentiate the functions with respect to $x$. $\sin(x^2 + 5)$
- Q2: Differentiate the functions with respect to $x$. $\cos(\sin x)$
- Q4: Differentiate the functions with respect to $x$. $\sec(\tan(\sqrt{x}))$
- Q5: Differentiate the functions with respect to $x$. $\frac{\sin(ax+b)}{\cos(cx+d)}$
- Q6: Differentiate the functions with respect to $x$. $\cos x^3 \cdot \sin^2(x^5)$
- Q7: Differentiate the functions with respect to $x$. $2\sqrt{\cot(x^2)}$
- Q8: Differentiate the functions with respect to $x$. $\cos(\sqrt{x})$
- Q9: Prove that the function $f$ given by $f(x) = |x - 1|$, $x \in R$ is not differentiable at $x = 1$.
- Q10: Prove that the greatest integer function defined by $f(x) = [x]$, $0 < x < 3$ is not differentiable at $x = 1$ and $x = 2$.
CBSE Solutions for Class 12 Mathematics Continuity and Differentiability
- Continuity and Differentiability Class 12 EXERCISE 5.1
- Continuity and Differentiability Class 12 EXERCISE 5.3
- Continuity and Differentiability Class 12 EXERCISE 5.4
- Continuity and Differentiability Class 12 EXERCISE 5.5
- Continuity and Differentiability Class 12 EXERCISE 5.6
- Continuity and Differentiability Class 12 EXERCISE 5.7
Chapters in CBSE - Class 12 Mathematics
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